For certain real numbers $a$, $b$, and $c$, the polynomial $g(x)=x^3+a{x}^2+2x+15$ has three distinct roots, and each root of $g(x)$ is also a root of the polynomial $f(x)=x^4+2x^3+b{x}^2+75x+c$. What is $f(2)$?