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

$$a ( a + 1 ) 2 ( a - 1 ) 2 ( a + 1 )$$

Evaluate

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

Solution Steps

Multiply a+1 and a+1 to get \left(a+1\right)^{2}.

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

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

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

Multiply 2 and 2 to get 4.

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

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

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

Use the distributive property to multiply a^{3}+2a^{2}+a by 4.

$$\left(4a^{3}+8a^{2}+4a\right)\left(a-1\right)$$

Apply the distributive property by multiplying each term of 4a^{3}+8a^{2}+4a by each term of a-1.

$$4a^{4}-4a^{3}+8a^{3}-8a^{2}+4a^{2}-4a$$

Combine -4a^{3} and 8a^{3} to get 4a^{3}.

$$4a^{4}+4a^{3}-8a^{2}+4a^{2}-4a$$

Combine -8a^{2} and 4a^{2} to get -4a^{2}.

$$4a^{4}+4a^{3}-4a^{2}-4a$$

Expand

$$4a^{4}+4a^{3}-4a^{2}-4a$$

Solution Steps

Multiply a+1 and a+1 to get \left(a+1\right)^{2}.

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

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

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

Multiply 2 and 2 to get 4.

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

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

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

Use the distributive property to multiply a^{3}+2a^{2}+a by 4.

$$\left(4a^{3}+8a^{2}+4a\right)\left(a-1\right)$$

Apply the distributive property by multiplying each term of 4a^{3}+8a^{2}+4a by each term of a-1.

$$4a^{4}-4a^{3}+8a^{3}-8a^{2}+4a^{2}-4a$$

Combine -4a^{3} and 8a^{3} to get 4a^{3}.

$$4a^{4}+4a^{3}-8a^{2}+4a^{2}-4a$$

Combine -8a^{2} and 4a^{2} to get -4a^{2}.

$$4a^{4}+4a^{3}-4a^{2}-4a$$