### Instruction: Suppose $f(x) = x^2,$ and $g(x)$ is a polynomial such that $f(g(x)) = 4x^2 + 4x + 1$. Enter all possible polynomials $g(x),$ separated by commas.### Response: I need to find $g(x)$ such that $f(g(x)) = 4x^2 + 4x + 1$.
I can start by writing $f(g(x))$ as $(g(x))^2$.
So I have $(g(x))^2 = 4x^2 + 4x + 1$.
I can try to factor the right-hand side and see if that helps.
I notice that $4x^2 + 4x + 1 = (2x + 1)^2$.
So I can rewrite the equation as $(g(x))^2 = (2x + 1)^2$.
Now I can take the square root of both sides and get $g(x) = \pm (2x + 1)$.
These are the two possible polynomials $g(x)$ that satisfy the given equation.
I can check them by plugging them back into $f(g(x))$ and see that they work.