Solve (3x+y)^{2}-(3x-y)^{2}+[(x+2y)^2+(x-2y)^2-8y^2]*2-(2x+3y)^2

$$( 3 x + y ) ^ { 2 } - ( 3 x - y ) ^ { 2 } + [ ( x + 2 y ) ^ { 2 } + ( x - 2 y ) ^ { 2 } - 8 y ^ { 2 } ] \cdot 2 - ( 2 x + 3 y ) ^ { 2 }$$

Evaluate

$$-9y^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+y\right)^{2}.

$$9x^{2}+6xy+y^{2}-\left(3x-y\right)^{2}+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-y\right)^{2}.

$$9x^{2}+6xy+y^{2}-\left(9x^{2}-6xy+y^{2}\right)+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

To find the opposite of 9x^{2}-6xy+y^{2}, find the opposite of each term.

$$9x^{2}+6xy+y^{2}-9x^{2}+6xy-y^{2}+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Combine 9x^{2} and -9x^{2} to get 0.

$$6xy+y^{2}+6xy-y^{2}+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Combine 6xy and 6xy to get 12xy.

$$12xy+y^{2}-y^{2}+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

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

$$12xy+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

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

$$12xy+\left(x^{2}+4xy+4y^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

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

$$12xy+\left(x^{2}+4xy+4y^{2}+x^{2}-4xy+4y^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

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

$$12xy+\left(2x^{2}+4xy+4y^{2}-4xy+4y^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Combine 4xy and -4xy to get 0.

$$12xy+\left(2x^{2}+4y^{2}+4y^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Combine 4y^{2} and 4y^{2} to get 8y^{2}.

$$12xy+\left(2x^{2}+8y^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Combine 8y^{2} and -8y^{2} to get 0.

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

Multiply 2 and 2 to get 4.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3y\right)^{2}.

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

To find the opposite of 4x^{2}+12xy+9y^{2}, find the opposite of each term.

$$12xy+4x^{2}-4x^{2}-12xy-9y^{2}$$

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

$$12xy-12xy-9y^{2}$$

Combine 12xy and -12xy to get 0.

$$-9y^{2}$$

Expand

$$-9y^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+y\right)^{2}.

$$9x^{2}+6xy+y^{2}-\left(3x-y\right)^{2}+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-y\right)^{2}.

$$9x^{2}+6xy+y^{2}-\left(9x^{2}-6xy+y^{2}\right)+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

To find the opposite of 9x^{2}-6xy+y^{2}, find the opposite of each term.

$$9x^{2}+6xy+y^{2}-9x^{2}+6xy-y^{2}+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Combine 9x^{2} and -9x^{2} to get 0.

$$6xy+y^{2}+6xy-y^{2}+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Combine 6xy and 6xy to get 12xy.

$$12xy+y^{2}-y^{2}+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

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

$$12xy+\left(\left(x+2y\right)^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

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

$$12xy+\left(x^{2}+4xy+4y^{2}+\left(x-2y\right)^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

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

$$12xy+\left(x^{2}+4xy+4y^{2}+x^{2}-4xy+4y^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

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

$$12xy+\left(2x^{2}+4xy+4y^{2}-4xy+4y^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Combine 4xy and -4xy to get 0.

$$12xy+\left(2x^{2}+4y^{2}+4y^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Combine 4y^{2} and 4y^{2} to get 8y^{2}.

$$12xy+\left(2x^{2}+8y^{2}-8y^{2}\right)\times 2-\left(2x+3y\right)^{2}$$

Combine 8y^{2} and -8y^{2} to get 0.

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

Multiply 2 and 2 to get 4.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3y\right)^{2}.

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

To find the opposite of 4x^{2}+12xy+9y^{2}, find the opposite of each term.

$$12xy+4x^{2}-4x^{2}-12xy-9y^{2}$$

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

$$12xy-12xy-9y^{2}$$

Combine 12xy and -12xy to get 0.

$$-9y^{2}$$