smc

Let $f(x)={\sum }_{k=0}^n{a}_k{x}^k$ be a polynomial satisfying the condition, so substituting it in, we find that the highest powers in each of the three expressions are, respectively, $a_n{x}^{2n}$, $a_n^2{x}^{2n}$, and $a_n^{n+1}{x}^{n^2}$. If polynomials are identically equal, each term must be equal, so we get $2n=n^2$ and $a_n=a_n^2=a_n^{n+1}$, so since $n\ge 1$, we must have $n=2$, and since $a_n\ne 0$, we have $a_n=1$. The given condition now becomes $x^4+b{x}^2+c\equiv (x^2+bx+c{)}^2$, so we must have $b=0$, or else the right-hand side would have a cubic term that the left-hand side does not. Thus we get $x^4+c\equiv (x^2+c{)}^2$, so we must have $c=0$, or else the right-hand side would have an $x^2$ term that the left-hand side does not. Thus the only possibility is $f(x)=x^2$, i.e. there is only $1$ solution, so the answer is $\boxed{1}$.