jmc

Seeing pairwise products, we consider $$(f+g+h+j{)}^2=f^2+g^2+h^2+j^2+2(fg+fh+fj+gh+gj+hj),$$ so $$fg+gh+hj+fj=\frac{(f+g+h+j{)}^2-f^2-g^2-h^2-j^2}{2}-(fh+gj).$$ Since the fraction on the right-hand side does not depend on how the values of $f$, $g$, $h$, and $j$ are assigned, we maximize $fg+gh+hj+fj$ by minimizing $fh+gj$. Checking the three distinct values for $fh+gj$, we find that $5\cdot 8+6\cdot 7=82$ is its minimum value. Therefore, the largest possible value of $fg+gh+hj+fj$ is $\frac{(5+6+7+8{)}^2-5^2-6^2-7^2-8^2}{2}-82=\boxed{169}$.