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

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

Evaluate

$$4ay+1$$

Solution Steps

Square a+1+2y.

$$4y^{2}+4ay+4y+a^{2}+2a+1-\left(a-1\right)\left(a+1\right)-\left(1+2y\right)^{2}-2a$$

Consider \left(a-1\right)\left(a+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$4y^{2}+4ay+4y+a^{2}+2a+1-\left(a^{2}-1\right)-\left(1+2y\right)^{2}-2a$$

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

$$4y^{2}+4ay+4y+a^{2}+2a+1-a^{2}+1-\left(1+2y\right)^{2}-2a$$

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

$$4y^{2}+4ay+4y+2a+1+1-\left(1+2y\right)^{2}-2a$$

Add 1 and 1 to get 2.

$$4y^{2}+4ay+4y+2a+2-\left(1+2y\right)^{2}-2a$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+2y\right)^{2}.

$$4y^{2}+4ay+4y+2a+2-\left(1+4y+4y^{2}\right)-2a$$

To find the opposite of 1+4y+4y^{2}, find the opposite of each term.

$$4y^{2}+4ay+4y+2a+2-1-4y-4y^{2}-2a$$

Subtract 1 from 2 to get 1.

$$4y^{2}+4ay+4y+2a+1-4y-4y^{2}-2a$$

Combine 4y and -4y to get 0.

$$4y^{2}+4ay+2a+1-4y^{2}-2a$$

Combine 4y^{2} and -4y^{2} to get 0.

$$4ay+2a+1-2a$$

Combine 2a and -2a to get 0.

$$4ay+1$$

Expand

$$4ay+1$$

Solution Steps

Square a+1+2y.

$$4y^{2}+4ay+4y+a^{2}+2a+1-\left(a-1\right)\left(a+1\right)-\left(1+2y\right)^{2}-2a$$

Consider \left(a-1\right)\left(a+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$4y^{2}+4ay+4y+a^{2}+2a+1-\left(a^{2}-1\right)-\left(1+2y\right)^{2}-2a$$

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

$$4y^{2}+4ay+4y+a^{2}+2a+1-a^{2}+1-\left(1+2y\right)^{2}-2a$$

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

$$4y^{2}+4ay+4y+2a+1+1-\left(1+2y\right)^{2}-2a$$

Add 1 and 1 to get 2.

$$4y^{2}+4ay+4y+2a+2-\left(1+2y\right)^{2}-2a$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+2y\right)^{2}.

$$4y^{2}+4ay+4y+2a+2-\left(1+4y+4y^{2}\right)-2a$$

To find the opposite of 1+4y+4y^{2}, find the opposite of each term.

$$4y^{2}+4ay+4y+2a+2-1-4y-4y^{2}-2a$$

Subtract 1 from 2 to get 1.

$$4y^{2}+4ay+4y+2a+1-4y-4y^{2}-2a$$

Combine 4y and -4y to get 0.

$$4y^{2}+4ay+2a+1-4y^{2}-2a$$

Combine 4y^{2} and -4y^{2} to get 0.

$$4ay+2a+1-2a$$

Combine 2a and -2a to get 0.

$$4ay+1$$