### 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 all polynomials $g(x)$ that satisfy the equation $f(g(x)) = 4x^2 + 4x + 1$.
Since $f(x) = x^2,$ I can rewrite the equation as $g(x)^2 = 4x^2 + 4x + 1$.
Now I need to think of how to factor the right-hand side, since it might help me find $g(x)$.
I notice that the right-hand side is a perfect square: it is equal to $(2x + 1)^2$.
So I can write the equation as $g(x)^2 = (2x + 1)^2$.
This means that either $g(x) = 2x + 1$ or $g(x) = -(2x + 1)$, since both of these functions squared give the same result.
These are the only two possible polynomials for $g(x)$, since any other polynomial would either have a different degree or different coefficients.