Okay, so I need to solve the inequality x² - 5x - 4 ≤ 10. Hmm, let me start by understanding what this means. The question is asking for all the values of x where the quadratic expression x² - 5x - 4 is less than or equal to 10. 

First step, maybe I should bring all terms to one side to make it easier to work with. If I subtract 10 from both sides, the inequality becomes x² - 5x - 4 - 10 ≤ 0. Simplifying that, it's x² - 5x - 14 ≤ 0. Okay, so now the inequality is x² - 5x - 14 ≤ 0. 

Now, I need to solve this quadratic inequality. I remember that to solve quadratic inequalities, it's helpful to find the roots of the corresponding quadratic equation first. So, let's consider the equation x² - 5x - 14 = 0. 

To find the roots, I can use the quadratic formula. The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a). Here, a = 1, b = -5, and c = -14. Plugging these into the formula, we get:

x = [5 ± √(25 - 4*1*(-14))]/2
x = [5 ± √(25 + 56)]/2
x = [5 ± √81]/2
√81 is 9, so this simplifies to:

x = [5 + 9]/2 = 14/2 = 7
x = [5 - 9]/2 = -4/2 = -2

So the roots are x = -2 and x = 7. These are the critical points that divide the real number line into intervals. The intervals are:

1. (-∞, -2)
2. (-2, 7)
3. (7, ∞)

Now, to determine where the quadratic expression x² - 5x - 14 is less than or equal to zero, I need to test each interval. Remember that the expression will be negative between the roots if the parabola opens upwards, which it does here because the coefficient of x² is positive.

So, the parabola opens upwards, meaning it has a minimum point. Therefore, the expression x² - 5x - 14 will be ≤ 0 between the roots -2 and 7. So the solution should be the interval [-2, 7].

But wait, let me verify this by testing a point in each interval.

First interval: Let's pick x = -3 (which is in (-∞, -2)). Plugging into the expression:

(-3)² -5*(-3) -14 = 9 +15 -14 = 10. Since 10 > 0, the expression is positive here.

Second interval: Let's take x = 0 (in (-2, 7)). Plugging in:

0² -5*0 -14 = -14. That's negative, so the expression is negative here.

Third interval: Let's take x = 8 (in (7, ∞)). Plugging in:

8² -5*8 -14 = 64 -40 -14 = 10. Positive again.

So, the expression is negative between -2 and 7, which confirms that the solution is the closed interval [-2, 7]. 

But just to be thorough, let me check the endpoints. At x = -2:

(-2)² -5*(-2) -14 = 4 +10 -14 = 0. Which is equal to 0, so it's included.

At x = 7:

7² -5*7 -14 = 49 -35 -14 = 0. Also included.

Therefore, the solution in interval notation is [-2,7].

I think that's all. Let me just recap the steps to make sure I didn't skip anything. First, moved 10 to the left side to form the inequality x² -5x -14 ≤ 0. Then found the roots using quadratic formula, which gave -2 and 7. Tested the intervals around the roots and found that the expression is negative between them. Confirmed by plugging in test points and checking the endpoints. Seems solid.
Therefore, the values of $x$ that satisfy the inequality are $x$ in the interval $[-2, 7]$.
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