### 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 a polynomial $g(x)$ that satisfies the equation $f(g(x)) = 4x^2 + 4x + 1$.
Since $f(x) = x^2,$ I can rewrite the equation as $(g(x))^2 = 4x^2 + 4x + 1$.
To solve for $g(x),$ I can take the square root of both sides, but I have to remember to include both the positive and negative roots.
So $g(x) = \pm \sqrt{4x^2 + 4x + 1}$.
Now I need to simplify the square root expression and see if it is a polynomial.
I can factor the quadratic inside the square root as $(2x + 1)^2.$
This means that $g(x) = \pm (2x + 1).$
Both of these are polynomials, so I have found all possible polynomials $g(x)$ that satisfy the equation.