Find the Area of y=(x+5)²-4: Quadratic Function Analysis

To find the positive area of the function $y = (x+5)^2-4$, follow these steps:

• Step 1: Identify when the function is greater than zero.
  We need $(x+5)^2 - 4 > 0$.
• Step 2: Find the roots of $(x+5)^2 - 4 = 0$.
  Solving, we set: $(x+5)^2 = 4$.
• Step 3: Solve for $(x+5)^2 = 4$.
  Take the square root on both sides: $x + 5 = \pm 2$.
  This gives: $x + 5 = 2$ or $x + 5 = -2$.
  Thus, $x = -3$ or $x = -7$.
• Step 4: Identify intervals.
  We need to look at intervals determined by the roots: $(-\infty, -7)$, $(-7, -3)$, and $(-3, \infty)$.
• Step 5: Determine where the function is positive by testing each interval:
  - For $x < -7$, choose $x = -8$, then $((-8)+5)^2 - 4 = 9 - 4 = 5$ (positive).
  - For $-7 < x < -3$, choose $x = -5$, then $((-5)+5)^2 - 4 = 0 - 4 = -4$ (negative).
  - For $x > -3$, choose $x = 0$, then $(5)^2 - 4 = 25 - 4 = 21$ (positive).

Therefore, the positive area of the function is for $x < -7$ and $-3 < x$.