jmc

Recall that if a function $f$ satisfies $f(x)=f(-x)$ for all real numbers $x$, then it is called an even function. Similarly, if $f(x)=-f(-x)$ for all real numbers $x$, then $f$ is called an odd function. Let's define $g(x)=x^4+x^2$ and $h(x)=5x$. Observe that $g(x)$ is even, $h(x)$ is odd, and $f(x)=g(x)+h(x)$. We have $$\begin{array}{rl}f(5)-f(-5)& =g(5)+h(5)-g(-5)-h(-5)\\ & =(g(5)-g(-5))+h(5)-h(-5)\\ & =0+2h(5)\\ & =2(5(5))\\ & =\boxed{50}.\end{array}$$