jmc

We use the distributive property to find $$\begin{array}{rl}& (2t^2-3t+2)(-3t^2+t-5)\\ & =2t^2(-3t^2+t-5)-3t(-3t^2+t-5)+2(-3t^2+t-5)\\ & =(-6t^4+2t^3-10t^2)+(9t^3-3t^2+15t)+(-6t^2+2t-10)\\ & =-6t^4+(2+9)t^3+(-10-3-6)t^2+(15+2)t-10\\ & =\boxed{-6t^4+11t^3-19t^2+17t-10}.\end{array}$$