Find the Descending Area of y=(x+5)²+2x: Quadratic Function Analysis

To determine where the function $y = (x+5)^2 + 2x$ is decreasing, we need to follow these steps:

• Step 1: Differentiate the Function
  The function is $y = (x+5)^2 + 2x$. First, we expand and simplify it:
  $y = (x+5)^2 + 2x = x^2 + 10x + 25 + 2x = x^2 + 12x + 25$.
  Now, compute the derivative:
  $y' = \frac{d}{dx} (x^2 + 12x + 25) = 2x + 12$.
• Step 2: Identify Intervals of Decrease
  Set the derivative less than zero to find where the function is decreasing:
  $2x + 12 < 0$.
  Solve for $x$:
  $2x < -12$
  $x < -6$.

Therefore, the function $y = (x+5)^2 + 2x$ is decreasing for $x < -6$.

The correct answer is $x < -6$.