I really need help with this exercise, I try by days. Use Heaviside in this exercise: $$f(t) = \begin{cases} 2t^2 + 4t +1 ,& 0 \leq t < 6 \\4t^2 -16t+50 ,& t \geq 6\end{cases}$$ for show L{f(t)} = $\dfrac{(s+2)^2}{s^3} + \dfrac{e^{-6s} (s+2)^2}{s^3}$
My solution: $f(t)$ = $2t^{2} + 4t +1 + [4t^2 -16t+50$ $-(2t^2 + 4t +1)]H(t-6)$
$2t^2 + 4t +1$ + $[2t^2 -20t +49]$ H(t-6)
$2t^2 + 4t +1$ + $[2(t-6)^2 +4(t-6)+1]$ H(t-6)
In the last step I used partial\ $[2t^2 -20t +49] = A(t-6)^2 + B(t-6)+C$