If $f(x)$ is symmetric about the line $x = -2$, find the values of $a$ and $b$.
Step-by-Step Solution
Key Concept: A function symmetric about $x = h$ satisfies $f(h+t) = f(h-t)$ for all $t$; this functional equation determines parameter constraints.
For $f(x)$ to be symmetric about $x = -2$, we need $f(-2+x) = f(-2-x)$. Expanding both sides and comparing coefficients, we get $a(2 + x)^2 + (2x + x)^2 + b(2 + x) + c = a(2 - x)^2 + (2 - x)^2 + b(2 - x) + c$. After simplification, the terms with odd powers of $x$ must vanish: $2ux + 2bx = 0$ leads to $b = 0$. The coefficient of $x^2$ and constant terms match when $a = 0$. Testing $a = 0, b = -4$ in the identity equation confirms it holds for all $x \in \mathbb{R}$.
Correct Answer: 2