<Question>
What is the maximum value of $4(x + 7)(2 - x)$, over all real numbers $x$?
</Question>

<think>
<Plan_1>
Okay, so I need to find the maximum value of the function 4(x + 7)(2 - x) for all real numbers x. Hmm, let's see. First, I should probably expand this expression to make it easier to work with. Let me start by expanding the parentheses.

None

<prompt_1.1> Expand the expression 4(x + 7)(2 - x) as per the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
So, (x + 7)(2 - x). Let me multiply those out. Using the distributive property, right? So x times 2 is 2x, and x times (-x) is -x². Then 7 times 2 is 14, and 7 times (-x) is -7x. So combining those terms: 2x - x² + 14 - 7x. Now, combine like terms. 2x - 7x is -5x, so the expression becomes -x² - 5x + 14. Then, don't forget the 4 multiplier outside the parentheses. So multiplying the entire thing by 4: 4*(-x² -5x +14) = -4x² -20x +56.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, so the quadratic is in standard form ax² + bx + c, where a is -4, b is -20, and c is 56.

None

<prompt_2.1> Identify the type of parabola and find the vertex of the quadratic function as per the above plan. </prompt_2.1>

So the vertex is at x = -b/(2a). Wait, but here b is -20, and a is -4. Let me plug those values in.

<prompt_2.2> Calculate the x-coordinate of the vertex using the formula -b/(2a) as per the above plan. </prompt_2.2>
-----

Alternatively, maybe I can use another method to verify. Let's see, maybe completing the square on the original quadratic. The original function after expanding was -4x² -20x +56. Let's factor out the coefficient of x² first.

<prompt_2.3> Use the completing the square method to verify the maximum value as per the above plan. </prompt_2.3>

Alternatively, maybe using calculus? Take the derivative and set it to zero. Let's try that.

<prompt_2.4> Use calculus to verify the maximum value by taking the derivative as per the above plan. </prompt_2.4>
</Plan_2>

<Execution_2>
<execution_2.1>
Since the coefficient of x² is negative (-4), the parabola opens downward, which means the vertex will be the maximum point. That makes sense because the problem is asking for the maximum value. So, the maximum value occurs at the vertex of the parabola.

Now, to find the vertex. For a quadratic in standard form, the x-coordinate of the vertex is given by -b/(2a). Let me confirm that. Yes, the formula is x = -b/(2a). So here, a = -4, b = -20. Plugging those in: x = -(-20)/(2*(-4)) = 20/(-8) = -2.5. Wait, so x is -2.5? Let me check that again.
</execution_2.1>

<execution_2.2>
So x = -(-20)/(2*(-4)) = 20/(-8) = -2.5. Wait, so x is -2.5. Hmm, but is that correct? Let me check my calculations again. The formula is -b/(2a). Here, a = -4, b = -20.

So substituting: -(-20) = 20, and 2a = 2*(-4) = -8. So 20 divided by -8 is indeed -2.5. So the x-coordinate of the vertex is -2.5.
</execution_2.2>

<execution_2.3>
So factor out -4: -4(x² + 5x) +56. Now, to complete the square inside the parentheses. The coefficient of x is 5, so take half of that, which is 2.5, square it, which is 6.25. So add and subtract 6.25 inside the parentheses. Wait, but since there's a factor of -4 outside, I need to be careful with the signs.

So, -4[(x² + 5x + 6.25 -6.25)] +56 = -4[(x + 2.5)^2 -6.25] +56. Distribute the -4: -4(x + 2.5)^2 +25 +56. Because -4*(-6.25) is +25. Then 25 +56 is 81. So the quadratic becomes -4(x + 2.5)^2 +81. Which again shows that the maximum value is 81 when the squared term is zero, i.e., when x = -2.5.
</execution_2.3>

<execution_2.4>
The function is f(x) = 4(x +7)(2 -x). Let's find f'(x). First, maybe expand it first to make differentiation easier, but I could also use the product rule. Let's expand first. As before, expanding gives -4x² -20x +56. Then the derivative f'(x) is -8x -20. Setting that equal to zero: -8x -20 = 0. Solving for x: -8x = 20 → x = -20/8 = -2.5. So x is -5/2. Then substitute back into the original function. f(-5/2) = 4*(-5/2 +7)(2 - (-5/2)). Let's compute each part. 

First, -5/2 +7 is the same as -2.5 +7 = 4.5, which is 9/2. Then, 2 - (-5/2) is 2 +5/2 = 9/2. So multiplying those together: 9/2 *9/2 =81/4. Then multiply by 4: 4*(81/4)=81. So again, we get 81. So that confirms it.
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.2:
Then plug that back into the original equation to find the maximum value. Let's do that.
First, compute f(-2.5) = -4*(-2.5)^2 -20*(-2.5) +56. Let's calculate each term step by step.

<prompt_3.1> Calculate the maximum value of the quadratic function at x = -2.5 as per the above plan. </prompt_3.1>
-----

Based on execution_2.1:
Alternatively, maybe there's another way to see this without expanding? Let me think.

<prompt_3.2> Explore alternative methods to find the maximum value without expanding as per the above plan. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
(-2.5)^2 is 6.25. So -4*6.25 is -25. Then, -20*(-2.5) is +50. Then +56. So adding those together: -25 +50 +56. That is 25 +56 = 81. So the maximum value is 81.
</execution_3.1>

<execution_3.2>
The original expression is 4(x +7)(2 -x). Maybe I can rewrite this in a way that makes it easier to find the maximum. Let's see. First, note that (x +7)(2 -x) can be written as -(x +7)(x -2). Which is a quadratic, but maybe there's a way to complete the square or use vertex formula without expanding. Alternatively, since it's a quadratic, the maximum occurs at the vertex, which is at x = (root1 + root2)/2. Wait, the roots of the quadratic (x +7)(2 -x) are x = -7 and x = 2. So the parabola has roots at -7 and 2, so the vertex is halfway between them. The midpoint between -7 and 2 is (-7 +2)/2 = (-5)/2 = -2.5. So x = -2.5, which is the same as before. Then, plugging back into the original expression: 4*(-2.5 +7)(2 -(-2.5)) = 4*(4.5)(4.5) = 4*(20.25) =81. So that's another way to get the same result without expanding the quadratic. That might have been a simpler approach. So, using the fact that for a quadratic with roots at r1 and r2, the vertex is at (r1 + r2)/2. That's a useful shortcut.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Wait, but let me verify that again to make sure I didn't make any calculation errors. Let's re-calculate f(-2.5).

<prompt_4.1> Re-calculate the maximum value at x = -2.5 to verify the result as per the above plan. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
First method: Original expression 4(x +7)(2 -x). Plugging x = -2.5:

x +7 = -2.5 +7 = 4.5

2 -x = 2 - (-2.5) = 2 +2.5 =4.5

Multiply them: 4.5 *4.5 = 20.25

Multiply by 4: 4*20.25 =81. That's the same result. So that's correct.

Alternatively, using the expanded form: -4x² -20x +56. Plugging x = -2.5:

First, x squared is (-2.5)^2 =6.25. So -4*6.25 = -25.

Then, -20*(-2.5) = +50.

Then, +56.

So total is -25 +50 +56 = 81.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the maximum value of $4(x + 7)(2 - x)$, over all real numbers $x$, is $\boxed{81}$.
</Final_answer>
</think>