Finding Negative Intervals in the Quadratic y=3x²+6x

We will work through the problem as follows:

• Step 1: Start by factoring the quadratic equation $y = 3x^2 + 6x$.
• Step 2: Solve for $x$ where $y = 0$.
• Step 3: Analyze the intervals between the roots to find where $y < 0$.

Let's proceed to the solution:
Step 1: Factor the quadratic expression:

The equation $y = 3x^2 + 6x$ can be factored by taking out the common factor:

$y = 3x(x + 2)$.

Step 2: Find the roots by setting $y = 0$:

$3x(x + 2) = 0$ gives the roots $x = 0$ and $x = -2$.

Step 3: Analyze the intervals determined by these roots:

• Interval $(-\infty, -2)$: Choose a test point like $x = -3$. Substituting into $3x(x + 2)$ gives a positive value.
• Interval $(-2, 0)$: Choose a test point like $x = -1$. Substituting into $3x(x + 2)$ gives a negative value.
• Interval $(0, \infty)$: Choose a test point like $x = 1$. Substituting into $3x(x + 2)$ gives a positive value.

Therefore, the function $y = 3x^2 + 6x$ is less than zero on the interval $-2 < x < 0$.

Therefore, the solution to the problem is $-2 < x < 0$.