This is problem 181 in Golan's linear algebra book. I have posted a proposed solution in the comments.
Problem: Let $V$, $W$, and $Y$ be vector spaces over a field $F$ and let $a\in Hom(V,W)$ and $b\in Hom(W,Y)$ satisfy the condition that $im(a)$ has a finitely generated complement in $W$ and $im(b)$ has a finitely generated complement in $Y$. Show that $im(b\circ a)$ has a finitely-generated complement in $Y$.