Solve (2x+3y)^2-(4x-y)^2

$$( 2 x + 3 y ) ^ { 2 } - ( 4 x - y ) ^ { 2 }$$

Evaluate

$$4\left(2y-x\right)\left(3x+y\right)$$

Solution Steps

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-y\right)^{2}.

$$4x^{2}+12xy+9y^{2}-\left(16x^{2}-8xy+y^{2}\right)$$

To find the opposite of 16x^{2}-8xy+y^{2}, find the opposite of each term.

$$4x^{2}+12xy+9y^{2}-16x^{2}+8xy-y^{2}$$

Combine 4x^{2} and -16x^{2} to get -12x^{2}.

$$-12x^{2}+12xy+9y^{2}+8xy-y^{2}$$

Combine 12xy and 8xy to get 20xy.

$$-12x^{2}+20xy+9y^{2}-y^{2}$$

Combine 9y^{2} and -y^{2} to get 8y^{2}.

$$-12x^{2}+20xy+8y^{2}$$

Expand

$$8y^{2}+20xy-12x^{2}$$

Solution Steps

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-y\right)^{2}.

$$4x^{2}+12xy+9y^{2}-\left(16x^{2}-8xy+y^{2}\right)$$

To find the opposite of 16x^{2}-8xy+y^{2}, find the opposite of each term.

$$4x^{2}+12xy+9y^{2}-16x^{2}+8xy-y^{2}$$

Combine 4x^{2} and -16x^{2} to get -12x^{2}.

$$-12x^{2}+12xy+9y^{2}+8xy-y^{2}$$

Combine 12xy and 8xy to get 20xy.

$$-12x^{2}+20xy+9y^{2}-y^{2}$$

Combine 9y^{2} and -y^{2} to get 8y^{2}.

$$-12x^{2}+20xy+8y^{2}$$