Prove that an equation has solution in R

Question

Let $f:\mathbb R\to \mathbb R$, $x\in\mathbb R$ and

$$f(x^2 + 3x + 1) = f^2(x) + 3f(x) + 1.$$

Prove that $f(x)=x$ has a solution $\in \mathbb R.$

Answer 1

Hint: Let $y$ be a solution to $x=x^2+3x+1$, then you get a quadratic equation in $f(y)$ which you can solve for $f(y)$.

Answer 2

$y^2+3y+1=y$
$y^2+2y+1$
$y = -1$

$f(y)=f^2(y)+3f(y)+1$
$f^2(y)+2f(y)+1=0$
$(f(y)+1)^2=0$
$f(y)=-1$

Therefore $f(x)=x$ has a solution $x=-1\in \mathbb{R}$