Solve the Quadratic Inequality: x² + 4x > 0

To solve the inequality $x^2 + 4x > 0$, we will:

• Step A: Find the roots of the equation $x^2 + 4x = 0$.
• Step B: Factor the quadratic to $x(x + 4) = 0$, giving roots $x = 0$ and $x = -4$.
• Step C: Use these roots to break the number line into intervals: $(-\infty, -4)$, $(-4, 0)$, and $(0, \infty)$.
• Step D: Test an arbitrary value from each interval in the inequality $x^2 + 4x > 0$.

Now, let's examine these intervals:

• For $(-\infty, -4)$, choose $x = -5$:
  $(-5)^2 + 4(-5) = 25 - 20 = 5 > 0$. This interval satisfies the inequality.
• For $(-4, 0)$, choose $x = -2$:
  $(-2)^2 + 4(-2) = 4 - 8 = -4 < 0$. This interval does not satisfy the inequality.
• For $(0, \infty)$, choose $x = 1$:
  $1^2 + 4(1) = 1 + 4 = 5 > 0$. This interval satisfies the inequality.

Therefore, the inequality $x^2 + 4x > 0$ holds true for the intervals $(-\infty, -4)$ and $(0, \infty)$.

Therefore, the solution to the inequality is $x < -4, 0 < x$.