# Laplace transform of following function, is this correct? 4H(t−3) * t^2

Laplace transform (Using tables) of following function, is this correct?

For my answer I used the the t-shift theorem (using tables) and I got
$4{e}^{-3s}.\frac{2}{{s}^{3}}$
I have a feeling this isn't correct.
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Zackary Mack
We need to write ${t}^{2}$ in terms of t−3 to use the Shifting Property or Laplace Transform definition (integral instead of tables).
We have
${t}^{2}=\left(\left(t-3\right)+3{\right)}^{2}=\left(t-3{\right)}^{2}+6\left(t-3\right)+9$
Hence, we can write
$4H\left(t-3\right){t}^{2}=4H\left(t-3\right)\left(\left(t-3{\right)}^{2}+6\left(t-3\right)+9\right)$
Using those three pieces, we end up with