Maximum number of real solutions of a polynomial

Question

Question statement: Find the maximum number of real solutions of $ax^n+x^2+bx+c$ where $a,b,c \in \mathbb{R}$ and $n$ is an even positive number.

Approach: The double derivative of the polynomial is

$f''(x)= n(n-1)ax^{n-2}+2$

For $a>0$, $f"(x)$ is never zero for real $x$. Hence there exists atmost 2 real solutions for $f(x)$.

Can't really proceed with the $a<0$ case.

Any better approach is invited.

PS: Sorry for the bad formatting. I am new here.

Answer 1

For $a<0$ and $n\ne 2$, $f''$ will have exactly two zeroes. Hence at most three roots for $f'$ and at most four roots for $f$. Four roots are achieved e.g. for $ -\frac1{10}x^4+x^2-1$.

If $a=0$ or $n=2$, we have a quadratic or lower polynomial, hence at most two roots. The only exception is: $n=2,a=-1,b=c=0$, which makes $f$ identically zero and hence uncountably many solutions.