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

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

Evaluate

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

Solution Steps

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

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

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

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

To find the opposite of a^{2}+2a+1-b^{2}, find the opposite of each term.

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

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

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

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

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

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

$$3a^{2}+4ab+2b^{2}-2a-1+a^{2}+2a+1$$

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

$$4a^{2}+4ab+2b^{2}-2a-1+2a+1$$

Combine -2a and 2a to get 0.

$$4a^{2}+4ab+2b^{2}-1+1$$

Add -1 and 1 to get 0.

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

Expand

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

Solution Steps

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

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

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

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

To find the opposite of a^{2}+2a+1-b^{2}, find the opposite of each term.

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

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

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

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

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

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

$$3a^{2}+4ab+2b^{2}-2a-1+a^{2}+2a+1$$

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

$$4a^{2}+4ab+2b^{2}-2a-1+2a+1$$

Combine -2a and 2a to get 0.

$$4a^{2}+4ab+2b^{2}-1+1$$

Add -1 and 1 to get 0.

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