Let V = { (ax^2+bx+c)e^{-x} |  a, b, & c  are real numbers }. 
Show that the derivative operator D takes V into V. That is show
that if f is in V then D(f) is in V. Using the basis { x^2e^{-x},
xe^{-x}, e^{-x} } find a matrix M representing D:V --> V. 
Show M is invertible and find M^{-1}. Use this matrix find write a
formula for Integral (ax^2+bx+c)e^{-x} dx (ignore the integration constant). 
If you want, verify it by using integration by parts!  

Note: If L:V --> V is invertible, as a function, then its matrix M is 
invertible and M^{-1} represents L^{-1}:V --> V. In this case D is 
one-to-one and onto. This is not typical.