Solve (a+1)(a^2+a+2)-(a^2-3a-5)(a-1)+3(1-2a^2)-5a

$$( a + 1 ) ( a ^ { 2 } + a + 2 ) - ( a ^ { 2 } - 3 a - 5 ) ( a - 1 ) + 3 ( 1 - 2 a ^ { 2 } ) - 5 a$$

Evaluate

$$0$$

Solution Steps

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

$$a^{3}+2a^{2}+3a+2-\left(a^{2}-3a-5\right)\left(a-1\right)+3\left(1-2a^{2}\right)-5a$$

Use the distributive property to multiply a^{2}-3a-5 by a-1 and combine like terms.

$$a^{3}+2a^{2}+3a+2-\left(a^{3}-4a^{2}-2a+5\right)+3\left(1-2a^{2}\right)-5a$$

To find the opposite of a^{3}-4a^{2}-2a+5, find the opposite of each term.

$$a^{3}+2a^{2}+3a+2-a^{3}+4a^{2}+2a-5+3\left(1-2a^{2}\right)-5a$$

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

$$2a^{2}+3a+2+4a^{2}+2a-5+3\left(1-2a^{2}\right)-5a$$

Combine 2a^{2} and 4a^{2} to get 6a^{2}.

$$6a^{2}+3a+2+2a-5+3\left(1-2a^{2}\right)-5a$$

Combine 3a and 2a to get 5a.

$$6a^{2}+5a+2-5+3\left(1-2a^{2}\right)-5a$$

Subtract 5 from 2 to get -3.

$$6a^{2}+5a-3+3\left(1-2a^{2}\right)-5a$$

Use the distributive property to multiply 3 by 1-2a^{2}.

$$6a^{2}+5a-3+3-6a^{2}-5a$$

Add -3 and 3 to get 0.

$$6a^{2}+5a-6a^{2}-5a$$

Combine 6a^{2} and -6a^{2} to get 0.

$$5a-5a$$

Combine 5a and -5a to get 0.

$$0$$

Factor

$$0$$