I want to get $L((t-4)^2u(t-4))$ I say this is a second shift with $g(t)=(t^2-4t)$ and my friend says "NO you are wrong, you are dumb!!!!!! $g(t)$ is MOST CERTAINLY equal to $t^2$"
Mine gives me $e^{-4s}(\frac{2}{s^3} - \frac{4}{s^2})$ his gives $e^{-4s}(\frac2{s^3})$
Who is correct?
Sorry, your friend has the right answer.
The second shifting theorem says that $$L(g(t-a)u(t-a))=e^{-as}G(s)\ .$$ To fit your problem into this format look at the $u(t-a)$ term: clearly you will need to take $a=4$. So to find $g$ you have $$g(t-4)=(t-4)^2$$ and substituting $t$ in place of $t-4$ gives $$g(t)=t^2\ .$$