Solve (a+b)^2-(3a+2)^2+2(2a+1)(2a-1)+4a=

$$( a + b ) ^ { 2 } - ( 3 a + 2 ) ^ { 2 } + 2 ( 2 a + 1 ) ( 2 a - 1 ) + 4 a =$$

Evaluate

$$2ab-8a+b^{2}-6$$

Solution Steps

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

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

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

$$a^{2}+2ab+b^{2}-\left(9a^{2}+12a+4\right)+2\left(2a+1\right)\left(2a-1\right)+4a$$

To find the opposite of 9a^{2}+12a+4, find the opposite of each term.

$$a^{2}+2ab+b^{2}-9a^{2}-12a-4+2\left(2a+1\right)\left(2a-1\right)+4a$$

Combine a^{2} and -9a^{2} to get -8a^{2}.

$$-8a^{2}+2ab+b^{2}-12a-4+2\left(2a+1\right)\left(2a-1\right)+4a$$

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

$$-8a^{2}+2ab+b^{2}-12a-4+\left(4a+2\right)\left(2a-1\right)+4a$$

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

$$-8a^{2}+2ab+b^{2}-12a-4+8a^{2}-2+4a$$

Combine -8a^{2} and 8a^{2} to get 0.

$$2ab+b^{2}-12a-4-2+4a$$

Subtract 2 from -4 to get -6.

$$2ab+b^{2}-12a-6+4a$$

Combine -12a and 4a to get -8a.

$$2ab+b^{2}-8a-6$$

Expand

$$2ab-8a+b^{2}-6$$

Solution Steps

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

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

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

$$a^{2}+2ab+b^{2}-\left(9a^{2}+12a+4\right)+2\left(2a+1\right)\left(2a-1\right)+4a$$

To find the opposite of 9a^{2}+12a+4, find the opposite of each term.

$$a^{2}+2ab+b^{2}-9a^{2}-12a-4+2\left(2a+1\right)\left(2a-1\right)+4a$$

Combine a^{2} and -9a^{2} to get -8a^{2}.

$$-8a^{2}+2ab+b^{2}-12a-4+2\left(2a+1\right)\left(2a-1\right)+4a$$

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

$$-8a^{2}+2ab+b^{2}-12a-4+\left(4a+2\right)\left(2a-1\right)+4a$$

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

$$-8a^{2}+2ab+b^{2}-12a-4+8a^{2}-2+4a$$

Combine -8a^{2} and 8a^{2} to get 0.

$$2ab+b^{2}-12a-4-2+4a$$

Subtract 2 from -4 to get -6.

$$2ab+b^{2}-12a-6+4a$$

Combine -12a and 4a to get -8a.

$$2ab+b^{2}-8a-6$$