jmc

Since $f(x)$ is a polynomial of degree $4$, its highest degree term is of the form $a{x}^4$. Substituting $x^3$ for $x$ shows that the highest degree term is $a(x^3{)}^4=a{x}^{12}$, which means that $f(x^3)$ has degree $12$. Similarly, $g(x^2)$ has degree $10$. Since the degree of the product of two polynomials is the sum of the degrees of the two polynomials, the degree of $f(x^3)\cdot g(x^2)$ is $12+10=\boxed{22}$.