Solve (a-1)(a+1)(a-4)+(a+2)^2*(a-1)+a(a+1)=

$$( a - 1 ) ( a + 1 ) ( a - 4 ) + ( a + 2 ) ^ { 2 } \cdot ( a - 1 ) + a ( a + 1 ) =$$

Evaluate

$$2a^{3}$$

Solution Steps

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

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

Use the distributive property to multiply a^{2}-1 by a-4.

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

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

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

Use the distributive property to multiply a^{2}+4a+4 by a-1 and combine like terms.

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

Combine a^{3} and a^{3} to get 2a^{3}.

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

Combine -4a^{2} and 3a^{2} to get -a^{2}.

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

Subtract 4 from 4 to get 0.

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

Use the distributive property to multiply a by a+1.

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

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

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

Combine -a and a to get 0.

$$2a^{3}$$

Expand

$$2a^{3}$$

Solution Steps

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

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

Use the distributive property to multiply a^{2}-1 by a-4.

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

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

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

Use the distributive property to multiply a^{2}+4a+4 by a-1 and combine like terms.

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

Combine a^{3} and a^{3} to get 2a^{3}.

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

Combine -4a^{2} and 3a^{2} to get -a^{2}.

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

Subtract 4 from 4 to get 0.

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

Use the distributive property to multiply a by a+1.

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

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

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

Combine -a and a to get 0.

$$2a^{3}$$