Find Decreasing Intervals for the Quadratic Function y = x² + 2x - 8

To solve this problem, we'll follow these steps:

• Step 1: Differentiate the given function.
• Step 2: Determine the critical point from the derivative.
• Step 3: Analyze the intervals based on the sign of the derivative.

Now, let's work through each step:
Step 1: Differentiate the function $y = x^2 + 2x - 8$. The derivative of $y$ with respect to $x$ is:
$\frac{dy}{dx} = 2x + 2$.

Step 2: Find the critical point by setting the derivative equal to zero:
$2x + 2 = 0$
$2x = -2$
$x = -1$.

Step 3: Analyze the intervals around $x = -1$:
For $x < -1$, pick a test point like $x = -2$, then $2(-2) + 2 = -4 + 2 = -2$, which is negative. Thus, the function is decreasing on $(-\infty, -1)$.
For $x > -1$, pick a test point like $x = 0$, then $2(0) + 2 = 2$, which is positive. Thus, the function is increasing on $(-1, \infty)$.

Therefore, the function is decreasing on the interval $x < -1$. Thus the correct choice is:

$x < -1$