jmc

$f(g(x))$ must have a degree of 8, because it will produce the term with the greatest exponent of the polynomial. Because $f(x)$ is a degree 4 polynomial, we can write that $f(x)=b{x}^4+c{x}^3+d{x}^2+ex+f$. The term with the greatest exponent in $f(g(x))$ comes from taking $b{x}^4$ or $b(g(x){)}^4$. Let $g(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_2{x}^2+a_1{x}^1+a_0$. Then, the highest degree term of $f(g(x))$ is $b(a_n{x}^n{)}^4$, which equals $b{a}_n^4{x}^{4n}$. Since the degree of $h$ is 8, we have $4n=8$, so $n=2$. Therefore, the degree of $g$ is $\boxed{2}$.