### 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 want to find all possible polynomials $g(x)$ that satisfy the equation $f(g(x)) = 4x^2 + 4x + 1$.
I know that $f(x) = x^2$, so I can plug in $g(x)$ for $x$ and get $f(g(x)) = (g(x))^2$.
Then, I can set $(g(x))^2$ equal to $4x^2 + 4x + 1$ and see if I can factor the right side.
I notice that $4x^2 + 4x + 1$ is a perfect square trinomial, since it can be written as $(2x + 1)^2$.
So, I have $(g(x))^2 = (2x + 1)^2$, and I can take the square root of both sides.
This gives me two possible cases: $g(x) = 2x + 1$ or $g(x) = -(2x + 1)$.
Both of these polynomials are valid, since they make the equation true.
Therefore, the answer is $g(x) = 2x + 1, -(2x + 1)$.