What is expand t^3(e^{-at}+1)^2 ?

The solution to expand t^3(e^{-at}+1)^2 is e^{-2ta}t^3+2e^{-ta}t^3+t^3

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expand t^3(e^{-at}+1)^2

expand

t^{3} (e^{- a t} + 1)^{2}

e^{- 2 t a} t^{3} + 2 e^{- t a} t^{3} + t^{3}

Solution steps

t^{3} (e^{- a t} + 1)^{2}

(e^{- a t} + 1)^{2} = e^{- 2 t a} + 2 e^{- t a} + 1

= t^{3} (e^{- 2 t a} + 2 e^{- t a} + 1)

Distribute parentheses

= t^{3} e^{- 2 t a} + t^{3} \cdot 2 e^{- t a} + t^{3} \cdot 1

= e^{- 2 t a} t^{3} + 2 e^{- t a} t^{3} + 1 \cdot t^{3}

Multiply:

1 \cdot t^{3} = t^{3}

= e^{- 2 t a} t^{3} + 2 e^{- t a} t^{3} + t^{3}

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What is expand t^3(e^{-at}+1)^2 ?
The solution to expand t^3(e^{-at}+1)^2 is e^{-2ta}t^3+2e^{-ta}t^3+t^3