Finding Solutions When x² + 4x is Positive

To solve the quadratic inequality $x^2 + 4x > 0$, we follow these steps:

• Step 1: Start by solving the quadratic equation $x^2 + 4x = 0$ to find its roots. Factoring, we get:

$x(x + 4) = 0$

This gives roots $x = 0$ and $x = -4$.

• Step 2: Use these roots to split the number line into intervals to test the sign of $x(x + 4)$ within each interval. The intervals are:
• Interval 1: $x < -4$
• Interval 2: $-4 < x < 0$
• Interval 3: $x > 0$

• Step 3: Choose a test point from each interval to determine the sign of $x(x + 4)$. For instance:
• Test point for Interval 1: $x = -5$. We have $(-5)((-5) + 4) = 5$, which is positive.
• Test point for Interval 2: $x = -2$. We have $(-2)((-2) + 4) = -4$, which is negative.
• Test point for Interval 3: $x = 1$. We have $1(1 + 4) = 5$, which is positive.

• Step 4: Compile the results. The inequality $x(x + 4) > 0$ holds for:
• $x < -4$ (Interval 1)
• $x > 0$ (Interval 3)

Thus the solution to the inequality $x^2 + 4x > 0$ is:

$x < -4$ or $x > 0$