Okay, so I need to figure out how many integer coordinates are strictly inside this rectangle. The rectangle has vertices at (5,4), (5,-4), (-5,4), (-5,-4). Hmm, let me visualize this first. The rectangle is centered at the origin, right? Because the x-coordinates go from -5 to 5, and the y-coordinates go from -4 to 4. Wait, actually, the vertices are (5,4), (5,-4), (-5,4), (-5,-4). So the sides are from x = -5 to x = 5 and y = -4 to y = 4. So the rectangle is 10 units wide along the x-axis and 8 units tall along the y-axis. 

But the question is asking for integer coordinates strictly inside the rectangle. So that means points where both the x and y coordinates are strictly inside the rectangle, not on the edges. So the edges are excluded. So for the x-coordinates, since the rectangle spans from x = -5 to x = 5, the strictly inside x-coordinates would be from x = -4 to x = 4. Similarly, for the y-coordinates, since it spans from y = -4 to y = 4, the strictly inside y-coordinates would be from y = -3 to y = 3. 

Wait, let me confirm. If the rectangle includes all points where x is between -5 and 5, inclusive, then the strictly inside points would be those where x is between -5 and 5, but not equal to -5 or 5. Similarly for y. So for x, the integers strictly inside would be -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 9 values. For y, the integers strictly inside would be -3, -2, -1, 0, 1, 2, 3. That's 7 values. 

Then, the total number of integer points strictly inside the rectangle would be the product of these two counts. So 9 multiplied by 7. Let me compute that: 9*7=63. So the answer should be 63. 

Wait, but let me check again. Maybe I made a mistake in counting the integers. For x, from -5 to 5 inclusive is 11 integers (-5,-4,-3,-2,-1,0,1,2,3,4,5). So strictly inside would exclude -5 and 5, leaving 9 integers. Similarly, for y, from -4 to 4 inclusive is 9 integers (-4,-3,-2,-1,0,1,2,3,4). Excluding -4 and 4, we have 7 integers. So 9*7=63. Yeah, that seems right. 

But let me think if there's another way to approach this. Maybe using the formula for the number of interior points. For a rectangle with integer vertices, the number of interior points can be calculated using the formula (width - 1) * (height - 1). Wait, but in this case, the rectangle's width is 10 (from -5 to 5) so the number of units along x is 10, but the number of interior points would be 10 - 2 = 8? Wait, no. Wait, maybe not. Let me clarify. 

The formula for the number of lattice points (points with integer coordinates) strictly inside a rectangle is (width - 2)*(height - 2). But wait, in this case, the width is 10 (from -5 to 5), so the number of interior points along x would be 10 - 2 = 8. Similarly, the height is 8 (from -4 to 4), so interior points along y would be 8 - 2 = 6. Then 8*6=48. But that contradicts the previous answer. So which one is correct? 

Hmm, maybe the formula is different. Wait, perhaps the formula is (number of interior x-coordinates) * (number of interior y-coordinates). So in this case, x ranges from -4 to 4, which is 9 integers, and y ranges from -3 to 3, which is 7 integers. So 9*7=63. So that's consistent with the first method. But why does the formula (width - 2)*(height - 2) give 48? Because width here is 10 units, height is 8 units. So (10 - 2)*(8 - 2)=8*6=48. But that's different from 63. So which is correct?

Wait, maybe the formula (width - 2)*(height - 2) applies when the sides are aligned with the grid. Wait, but in our case, the rectangle is from (-5, -4) to (5,4). So the width is 10 (from x=-5 to x=5) and the height is 8 (from y=-4 to y=4). So if we subtract 2 from each dimension, it's 8 and 6, giving 48. But why is there a discrepancy?

Wait, maybe the formula is (length - 1)*(width - 1), but that doesn't fit here. Let me check with a simple example. Suppose a rectangle from (0,0) to (2,2). The strictly interior points would be (1,1), which is 1 point. Using the first method: x from 1 to 1 (since excluding 0 and 2), y from 1 to 1. So 1*1=1. Using the formula (2-2)*(2-2)=0*0=0. That's wrong. So the formula (width - 2)*(height - 2) doesn't work here. So maybe the first method is correct.

Wait, in the example from (0,0) to (2,2), the width is 2, height is 2. The strictly interior points are 1. So according to the first method, x from 0+1 to 2-1=1, so 1 value. Similarly for y. So 1*1=1. Correct. So maybe the correct formula is (x_end - x_start - 1)*(y_end - y_start -1). Wait, in this case, x_end is 5, x_start is -5. So 5 - (-5) -1 = 10 -1=9. Similarly, y_end is 4, y_start is -4. 4 - (-4) -1=8 -1=7. Then 9*7=63. Which matches the first method. 

So the general formula for the number of interior points is (x_end - x_start -1) * (y_end - y_start -1). So in this problem, x_end=5, x_start=-5, so 5 - (-5) -1=9. y_end=4, y_start=-4, so 4 - (-4) -1=7. 9*7=63. So that's the answer. 

But let me check another example to confirm. Suppose a rectangle from (1,1) to (3,3). The interior points would be (2,2). So 1 point. Using the formula: x_end=3, x_start=1. 3 -1 -1=1. y_end=3, y_start=1. 3 -1 -1=1. 1*1=1. Correct. 

Another example: rectangle from (0,0) to (3,2). The interior points would be x=1,2 and y=1. So 2*1=2 points. Using the formula: 3-0-1=2, 2-0-1=1. 2*1=2. Correct. 

So yes, the formula works. Therefore, applying it here: 9*7=63. So the answer is 63. 

Alternatively, maybe I can count the number of integer x's strictly inside and multiply by the number of integer y's strictly inside. For x, strictly inside the rectangle means x > -5 and x < 5. Since x must be integer, x can be -4, -3, ..., 4. That's 9 values. For y, strictly inside means y > -4 and y < 4. So y can be -3, -2, ..., 3. That's 7 values. So total points 9*7=63. 

Yes, that's another way to see it. So both methods lead to the same answer. Therefore, I can be confident that the answer is 63. 

But just to make sure, let me try to list some points and see. For example, x ranges from -4 to 4. Let's take x=-4. Then y can be from -3 to 3. So points like (-4,-3), (-4,-2), ..., (-4,3). That's 7 points. Similarly for x=-3, same y's. So 7 points for each x. There are 9 x's, so 9*7=63. 

Alternatively, think of the rectangle as a grid. The rectangle spans 10 units in x (from -5 to 5) and 8 units in y (from -4 to 4). The number of lattice points inside would be (10 - 2)*(8 - 2)=8*6=48. Wait, but that's conflicting with the previous result. Hmm, why is there a discrepancy?

Wait, maybe the formula (width - 2)*(height - 2) is not applicable here. Because in our case, the rectangle is not aligned with the grid in a way that the interior points form a grid. Wait, actually, the rectangle is axis-aligned, so the interior points should indeed form a grid. So if the rectangle has width W and height H, then the number of interior points is (W - 2)*(H - 2). But in our case, W is 10 (from -5 to 5 inclusive) and H is 8 (from -4 to 4 inclusive). So (10 - 2)*(8 - 2)=8*6=48. But earlier calculation gave 63. Which one is correct?

Wait, maybe the formula is (W - 1)*(H - 1). Wait, in our first approach, the number of x's is 9 and y's is 7, so 9*7=63. But 9 is 10 -1, and 7 is 8 -1. So maybe the formula is (W -1)*(H -1). But in that case, W is the total width, which is 10, so 10 -1=9, and H=8, so 8 -1=7. So 9*7=63. That matches. So maybe the formula is (width -1)*(height -1). But why does the other formula give 48?

Wait, confusion arises because there are two different ways to define width and height. If the width is the number of integer points along the x-axis, then from x=-5 to x=5 inclusive, there are 11 integer points. So the number of interior points would be 11 - 2 = 9. Similarly, height from y=-4 to y=4 is 9 integer points, so interior points 9 - 2 =7. Then total interior points 9*7=63. 

Alternatively, if the width is the difference between x coordinates, which is 5 - (-5) =10, then the number of interior points is 10 - 2=8. Similarly, height is 4 - (-4)=8, so interior points 8 -2=6. Then 8*6=48. 

So which is correct? The answer depends on whether we consider the width as the count of integer points or the difference between the coordinates. 

But in the problem statement, it says "integer coordinates strictly inside the rectangular region". So the strictly inside points are those that are not on the boundary. The boundary of the rectangle includes all points where x is in {-5,5} or y is in {-4,4}. So strictly inside points have x and y not in those sets. Therefore, x must be between -5 and 5, but not equal to -5 or 5, and y must be between -4 and 4, but not equal to -4 or 4. So that's x in {-4,-3,-2,-1,0,1,2,3,4} (9 values) and y in {-3,-2,-1,0,1,2,3} (7 values). Therefore, 9*7=63. 

Therefore, the correct answer is 63. The formula (width -1)*(height -1) where width is the count of integer points along x and height is the count along y is correct here. The confusion with the other formula arises when width and height are defined as the differences between coordinates, which would be 10 and 8, leading to 8*6=48. But that's incorrect because the width in terms of integer points is 11 (from -5 to 5 inclusive), so interior points 9. Similarly, height 9 (from -4 to 4 inclusive), interior points 7. 

Therefore, the answer is 63. 

Another way to think about it: The rectangle has vertices at (5,4), (5,-4), etc. So the sides are x=5, x=-5, y=4, y=-4. The strictly inside region is where -5 < x <5 and -4 < y <4. Since x and y must be integers, the integers strictly inside are x from -4 to 4 inclusive, which is 9 numbers, and y from -3 to 3 inclusive, which is 7 numbers. Multiply them to get 63. 

Yes, that seems solid. I think I've confirmed it multiple ways. So the answer is 63.
Therefore, the number of integer coordinates strictly inside the rectangular region is $\boxed{63}$.
</think>