What is laplacetransform (t+2)^2e^t ?

The answer to laplacetransform (t+2)^2e^t is (4s^2-4s+2)/((s-1)^3)

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laplacetransform (t+2)^2e^t

laplace transform

(t + 2)^{2} e^{t}

\frac{4 s ^{2} - 4 s + 2}{( s - 1 ) ^{3}}

Solution steps

L {e^{t} (t + 2)^{2}}

Apply transform rule:

L {f (t)} = F (s)

then

L {e^{a t} f (t)} = F (s - a)

For

(t + 2)^{2} e^{t} : f (t) = (t + 2)^{2}, a = 1

= L {(t + 2)^{2}} (s - 1)

L {(t + 2)^{2}} : \frac{2}{s ^{3}} + \frac{4}{s ^{2}} + \frac{4}{s}

= \frac{2}{( s - 1 ) ^{3}} + \frac{4}{( s - 1 ) ^{2}} + \frac{4}{( s - 1 )}

= \frac{2}{( s - 1 ) ^{3}} + \frac{4}{( s - 1 ) ^{2}} + \frac{4}{s - 1}

Simplify

\frac{2}{( s - 1 ) ^{3}} + \frac{4}{( s - 1 ) ^{2}} + \frac{4}{s - 1} : \frac{4 s ^{2} - 4 s + 2}{( s - 1 ) ^{3}}

= \frac{4 s ^{2} - 4 s + 2}{( s - 1 ) ^{3}}

What is laplacetransform (t+2)^2e^t ?
The answer to laplacetransform (t+2)^2e^t is (4s^2-4s+2)/((s-1)^3)