Solve x+1+(2x+y+3)^2-(2x+y)^2-2(3y)

$$x + 1 + ( 2 x + y + 3 ) ^ { 2 } - ( 2 x + y ) ^ { 2 } - 2 ( 3 y )$$

Evaluate

$$13x+10$$

Solution Steps

Square 2x+y+3.

$$x+1+4x^{2}+4xy+12x+y^{2}+6y+9-\left(2x+y\right)^{2}-2\times 3y$$

Combine x and 12x to get 13x.

$$13x+1+4x^{2}+4xy+y^{2}+6y+9-\left(2x+y\right)^{2}-2\times 3y$$

Add 1 and 9 to get 10.

$$13x+10+4x^{2}+4xy+y^{2}+6y-\left(2x+y\right)^{2}-2\times 3y$$

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

$$13x+10+4x^{2}+4xy+y^{2}+6y-\left(4x^{2}+4xy+y^{2}\right)-2\times 3y$$

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

$$13x+10+4x^{2}+4xy+y^{2}+6y-4x^{2}-4xy-y^{2}-2\times 3y$$

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

$$13x+10+4xy+y^{2}+6y-4xy-y^{2}-2\times 3y$$

Combine 4xy and -4xy to get 0.

$$13x+10+y^{2}+6y-y^{2}-2\times 3y$$

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

$$13x+10+6y-2\times 3y$$

Multiply 2 and 3 to get 6.

$$13x+10+6y-6y$$

Combine 6y and -6y to get 0.

$$13x+10$$

Expand

$$13x+10$$

Solution Steps

Square 2x+y+3.

$$x+1+4x^{2}+4xy+12x+y^{2}+6y+9-\left(2x+y\right)^{2}-2\times 3y$$

Combine x and 12x to get 13x.

$$13x+1+4x^{2}+4xy+y^{2}+6y+9-\left(2x+y\right)^{2}-2\times 3y$$

Add 1 and 9 to get 10.

$$13x+10+4x^{2}+4xy+y^{2}+6y-\left(2x+y\right)^{2}-2\times 3y$$

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

$$13x+10+4x^{2}+4xy+y^{2}+6y-\left(4x^{2}+4xy+y^{2}\right)-2\times 3y$$

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

$$13x+10+4x^{2}+4xy+y^{2}+6y-4x^{2}-4xy-y^{2}-2\times 3y$$

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

$$13x+10+4xy+y^{2}+6y-4xy-y^{2}-2\times 3y$$

Combine 4xy and -4xy to get 0.

$$13x+10+y^{2}+6y-y^{2}-2\times 3y$$

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

$$13x+10+6y-2\times 3y$$

Multiply 2 and 3 to get 6.

$$13x+10+6y-6y$$

Combine 6y and -6y to get 0.

$$13x+10$$