Solve the Quadratic Inequality: 3x² + 6x > 0

To solve the inequality $3x^2 + 6x > 0$, we first factor the quadratic equation.

• Step 1: Factor the quadratic expression:
  $3x^2 + 6x = 3x(x + 2)$.
• Step 2: Find the roots by setting the factored expression equal to zero:
  - $3x(x + 2) = 0$ gives $x = 0$ and $x = -2$ as roots.
• Step 3: Determine the intervals dictated by the roots on a number line:
  - The critical points divide the number line into three intervals: $(-\infty, -2)$, $(-2, 0)$, and $(0, \infty)$.
• Step 4: Analyze the sign of $3x(x + 2)$ in each interval:
• Interval $(-\infty, -2)$: Pick a test point like $x = -3$.
  $3(-3)((-3) + 2) = 3(-3)(-1) = 9 > 0$. Hence, $f(x) > 0$.
• Interval $(-2, 0)$: Pick a test point like $x = -1$.
  $3(-1)((-1) + 2) = 3(-1)(1) = -3 < 0$. Hence, $f(x) < 0$.
• Interval $(0, \infty)$: Pick a test point like $x = 1$.
  $3(1)((1) + 2) = 3(1)(3) = 9 > 0$. Hence, $f(x) > 0$.

Therefore, the solution to the inequality $3x^2 + 6x > 0$ is:
$x > 0$ or $x < -2$.

The correct choice from the given options is .

Thus, when $x > 0$ or $x < -2$, the function $f(x) = 3x^2 + 6x$ is greater than zero.