Find Decreasing Intervals for y = (3+x)(x-7): Quadratic Function Analysis

To solve this problem, we will follow these steps:

• Step 1: Expand the function $y = (3+x)(x-7)$ into standard form.
• Step 2: Differentiate the expanded quadratic function with respect to $x$.
• Step 3: Set the derivative less than zero to find where the function is decreasing.

Now, let's work through each step:

Step 1: Expand the function:

$y = (3+x)(x-7) = x^2 - 7x + 3x - 21 = x^2 - 4x - 21$.

Step 2: Take the derivative of $y = x^2 - 4x - 21$:

$y' = \frac{d}{dx}[x^2 - 4x - 21] = 2x - 4$.

Step 3: Set the derivative less than zero to determine where the function is decreasing:

$2x - 4 < 0$.

Solve the inequality:

$2x < 4$

$x < 2$.

This indicates that the function is decreasing for $x < 2$.

Upon checking the given choices, we find that the correct answer is $x < 2$.

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