Determining X: When is x² + 4x Less Than Zero?

To solve the inequality $x^2 + 4x < 0$, we first need to determine the roots of the quadratic equation $x^2 + 4x = 0$.

Step 1: Find roots of the equation:

Factor the quadratic expression: $x(x + 4) = 0$.

Setting each factor to zero gives us the roots:

• $x = 0$
• $x + 4 = 0 \Rightarrow x = -4$

Step 2: Analyze intervals between the roots:

The roots divide the real number line into intervals: $(-\infty, -4)$, $(-4, 0)$, and $(0, \infty)$.

Step 3: Test the sign of $f(x) = x^2 + 4x$ in each interval:

• For $x \in (-\infty, -4)$, pick $x = -5$: $f(-5) = (-5)^2 + 4(-5) = 25 - 20 = 5$ (positive).
• For $x \in (-4, 0)$, pick $x = -2$: $f(-2) = (-2)^2 + 4(-2) = 4 - 8 = -4$ (negative).
• For $x \in (0, \infty)$, pick $x = 1$: $f(1) = 1^2 + 4(1) = 1 + 4 = 5$ (positive).

Step 4: Conclusion:

The function $f(x) = x^2 + 4x$ is negative in the interval $(-4, 0)$, specifically $-4 < x < 0$.

Therefore, the values of $x$ for which $f(x) < 0$ are in the interval $-4 < x < 0$.