Solve (2a+b)^3-(2a-b)^3+2b(a+b)(a-b)

$$( 2 a + b ) ^ { 3 } - ( 2 a - b ) ^ { 3 } + 2 b ( a + b ) ( a - b )$$

Evaluate

$$26ba^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{3}=p^{3}+3p^{2}q+3pq^{2}+q^{3} to expand \left(2a+b\right)^{3}.

$$8a^{3}+12a^{2}b+6ab^{2}+b^{3}-\left(2a-b\right)^{3}+2b\left(a+b\right)\left(a-b\right)$$

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(2a-b\right)^{3}.

$$8a^{3}+12a^{2}b+6ab^{2}+b^{3}-\left(8a^{3}-12a^{2}b+6ab^{2}-b^{3}\right)+2b\left(a+b\right)\left(a-b\right)$$

To find the opposite of 8a^{3}-12a^{2}b+6ab^{2}-b^{3}, find the opposite of each term.

$$8a^{3}+12a^{2}b+6ab^{2}+b^{3}-8a^{3}+12a^{2}b-6ab^{2}+b^{3}+2b\left(a+b\right)\left(a-b\right)$$

Combine 8a^{3} and -8a^{3} to get 0.

$$12a^{2}b+6ab^{2}+b^{3}+12a^{2}b-6ab^{2}+b^{3}+2b\left(a+b\right)\left(a-b\right)$$

Combine 12a^{2}b and 12a^{2}b to get 24a^{2}b.

$$24a^{2}b+6ab^{2}+b^{3}-6ab^{2}+b^{3}+2b\left(a+b\right)\left(a-b\right)$$

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

$$24a^{2}b+b^{3}+b^{3}+2b\left(a+b\right)\left(a-b\right)$$

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

$$24a^{2}b+2b^{3}+2b\left(a+b\right)\left(a-b\right)$$

Use the distributive property to multiply 2b by a+b.

$$24a^{2}b+2b^{3}+\left(2ba+2b^{2}\right)\left(a-b\right)$$

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

$$24a^{2}b+2b^{3}+2ba^{2}-2b^{3}$$

Combine 24a^{2}b and 2ba^{2} to get 26a^{2}b.

$$26a^{2}b+2b^{3}-2b^{3}$$

Combine 2b^{3} and -2b^{3} to get 0.

$$26a^{2}b$$

Expand

$$26ba^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{3}=p^{3}+3p^{2}q+3pq^{2}+q^{3} to expand \left(2a+b\right)^{3}.

$$8a^{3}+12a^{2}b+6ab^{2}+b^{3}-\left(2a-b\right)^{3}+2b\left(a+b\right)\left(a-b\right)$$

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(2a-b\right)^{3}.

$$8a^{3}+12a^{2}b+6ab^{2}+b^{3}-\left(8a^{3}-12a^{2}b+6ab^{2}-b^{3}\right)+2b\left(a+b\right)\left(a-b\right)$$

To find the opposite of 8a^{3}-12a^{2}b+6ab^{2}-b^{3}, find the opposite of each term.

$$8a^{3}+12a^{2}b+6ab^{2}+b^{3}-8a^{3}+12a^{2}b-6ab^{2}+b^{3}+2b\left(a+b\right)\left(a-b\right)$$

Combine 8a^{3} and -8a^{3} to get 0.

$$12a^{2}b+6ab^{2}+b^{3}+12a^{2}b-6ab^{2}+b^{3}+2b\left(a+b\right)\left(a-b\right)$$

Combine 12a^{2}b and 12a^{2}b to get 24a^{2}b.

$$24a^{2}b+6ab^{2}+b^{3}-6ab^{2}+b^{3}+2b\left(a+b\right)\left(a-b\right)$$

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

$$24a^{2}b+b^{3}+b^{3}+2b\left(a+b\right)\left(a-b\right)$$

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

$$24a^{2}b+2b^{3}+2b\left(a+b\right)\left(a-b\right)$$

Use the distributive property to multiply 2b by a+b.

$$24a^{2}b+2b^{3}+\left(2ba+2b^{2}\right)\left(a-b\right)$$

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

$$24a^{2}b+2b^{3}+2ba^{2}-2b^{3}$$

Combine 24a^{2}b and 2ba^{2} to get 26a^{2}b.

$$26a^{2}b+2b^{3}-2b^{3}$$

Combine 2b^{3} and -2b^{3} to get 0.

$$26a^{2}b$$