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

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

Evaluate

$$4a^{2}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine 2ab and -2ab to get 0.

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

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

$$4a^{2}$$

Expand

$$4a^{2}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine 2ab and -2ab to get 0.

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

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

$$4a^{2}$$