jmc

Using the distributive property, we can expand this to get $$\begin{array}{rl}(x^{22}& -3x^5+x^{-2}-7)\cdot (5x^4)\\ & =(x^{22})(5x^4)+(-3x^5)(5x^4)+(x^{-2})(5x^4)-7(5x^4)\\ & =5x^{26}-15x^9+5x^2-35x^4\\ & =\boxed{5x^{26}-15x^9-35x^4+5x^2}.\end{array}$$