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

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

Evaluate

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

Solution Steps

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

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

To find the opposite of a-b, find the opposite of each term.

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

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

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

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

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

Combine b^{2} and 3b^{2} to get 4b^{2}.

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

Combine -a and -4a to get -5a.

$$4a^{2}+2ab+4b^{2}-5a+b+4b$$

Combine b and 4b to get 5b.

$$4a^{2}+2ab+4b^{2}-5a+5b$$

Expand

$$4a^{2}+2ab-5a+4b^{2}+5b$$

Solution Steps

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

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

To find the opposite of a-b, find the opposite of each term.

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

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

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

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

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

Combine b^{2} and 3b^{2} to get 4b^{2}.

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

Combine -a and -4a to get -5a.

$$4a^{2}+2ab+4b^{2}-5a+b+4b$$

Combine b and 4b to get 5b.

$$4a^{2}+2ab+4b^{2}-5a+5b$$