Solve (a-1)^2(a+2)+(1-a)(a+1)(a-2)=

$$( a - 1 ) ^ { 2 } ( a + 2 ) + ( 1 - a ) ( a + 1 ) ( a - 2 ) =$$

Evaluate

$$2a\left(a-1\right)$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-1\right)^{2}.

$$\left(a^{2}-2a+1\right)\left(a+2\right)+\left(1-a\right)\left(a+1\right)\left(a-2\right)$$

Use the distributive property to multiply a^{2}-2a+1 by a+2 and combine like terms.

$$a^{3}-3a+2+\left(1-a\right)\left(a+1\right)\left(a-2\right)$$

Use the distributive property to multiply 1-a by a+1 and combine like terms.

$$a^{3}-3a+2+\left(1-a^{2}\right)\left(a-2\right)$$

Use the distributive property to multiply 1-a^{2} by a-2.

$$a^{3}-3a+2+a-2-a^{3}+2a^{2}$$

Combine -3a and a to get -2a.

$$a^{3}-2a+2-2-a^{3}+2a^{2}$$

Subtract 2 from 2 to get 0.

$$a^{3}-2a-a^{3}+2a^{2}$$

Combine a^{3} and -a^{3} to get 0.

$$-2a+2a^{2}$$

Expand

$$2a^{2}-2a$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-1\right)^{2}.

$$\left(a^{2}-2a+1\right)\left(a+2\right)+\left(1-a\right)\left(a+1\right)\left(a-2\right)$$

Use the distributive property to multiply a^{2}-2a+1 by a+2 and combine like terms.

$$a^{3}-3a+2+\left(1-a\right)\left(a+1\right)\left(a-2\right)$$

Use the distributive property to multiply 1-a by a+1 and combine like terms.

$$a^{3}-3a+2+\left(1-a^{2}\right)\left(a-2\right)$$

Use the distributive property to multiply 1-a^{2} by a-2.

$$a^{3}-3a+2+a-2-a^{3}+2a^{2}$$

Combine -3a and a to get -2a.

$$a^{3}-2a+2-2-a^{3}+2a^{2}$$

Subtract 2 from 2 to get 0.

$$a^{3}-2a-a^{3}+2a^{2}$$

Combine a^{3} and -a^{3} to get 0.

$$-2a+2a^{2}$$