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

The solution to expand (e^t+at-2)(t^3-2t^2+1) is e^tt^3-2e^tt^2+e^t+t^4a-2t^3a+ta-2t^3+4t^2-2

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

expand

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

e^{t} t^{3} - 2 e^{t} t^{2} + e^{t} + t^{4} a - 2 t^{3} a + t a - 2 t^{3} + 4 t^{2} - 2

Solution steps

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

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

Simplify

e^{t} t^{3} - 2 e^{t} t^{2} + 1 \cdot e^{t} + t^{3} t a - 2 t^{2} t a + 1 \cdot t a - 2 t^{3} + 2 \cdot 2 t^{2} - 2 \cdot 1 : e^{t} t^{3} - 2 e^{t} t^{2} + e^{t} + t^{4} a - 2 t^{3} a + t a - 2 t^{3} + 4 t^{2} - 2

= e^{t} t^{3} - 2 e^{t} t^{2} + e^{t} + t^{4} a - 2 t^{3} a + t a - 2 t^{3} + 4 t^{2} - 2

\frac{5 s ^{2} + 44}{( s ^{2} + 44 ) ( s + 2 )}

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

\frac{( a ^{2} + 9 ) ( a ^{2} - 3 ) 3}{4 a ^{2}}

\frac{1}{s ( s ^{2} + A ^{2} )}

4 + f (2 t - 7) H (t - 2)

What is expand (e^t+at-2)(t^3-2t^2+1) ?
The solution to expand (e^t+at-2)(t^3-2t^2+1) is e^tt^3-2e^tt^2+e^t+t^4a-2t^3a+ta-2t^3+4t^2-2