Solve the Quadratic Inequality: -x² - 10x > 0

To solve the inequality $-x^2 - 10x > 0$, follow these steps:

• Rewrite the inequality: $-x^2 - 10x > 0$.
• Factor the quadratic expression: $-x(x + 10) > 0$.
• Identify the roots of the equation $-x(x + 10) = 0$. The roots are $x = 0$ and $x = -10$.
• These roots divide the number line into three intervals: $(-\infty, -10)$, $(-10, 0)$, and $(0, \infty)$.
• Test a point from each interval to determine where the inequality holds:
• For the interval $(-\infty, -10)$, test $x = -11$:
  Substitute $x = -11$ into $-x(-x - 10) = -(-11)(-11 + 10) = -11 \cdot (-1) = 11$, which is positive, but we need it to be positive; hence it does not satisfy the inequality.
• For the interval $(-10, 0)$, test $x = -5$:
  Substitute $x = -5$ into $-x(-x - 10) = -(-5)(-5 + 10) = 5 \cdot 5 = 25$, which is positive. This interval satisfies the inequality.
• For the interval $(0, \infty)$, test $x = 1$:
  Substitute $x = 1$ into $-x(x + 10) = -(1)(1 + 10) = -11$, which is negative; hence it does not satisfy the inequality.

The solution to the inequality $-x^2 - 10x > 0$ is the interval $-10 < x < 0$, which matches choice 2.