### 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 know that $f(x) = x^2,$ so if I plug in $g(x)$ instead of $x,$ I get $f(g(x)) = (g(x))^2$.
This means that $(g(x))^2$ must be equal to $4x^2 + 4x + 1$.
I notice that $4x^2 + 4x + 1$ is a perfect square, namely $(2x + 1)^2$.
So I can write $(g(x))^2 = (2x + 1)^2$, and then take the square root of both sides.
This gives me two possibilities: either $g(x) = 2x + 1,$ or $g(x) = -(2x + 1)$.
These are both polynomials, so they are both valid answers.
Therefore, the possible polynomials $g(x)$ are 2x + 1, -(2x + 1).