Solve (3x-2y)^2-(2x+3y)^2-(15x+y)(x-5y)]

$$( 3 x - 2 y ) ^ { 2 } - ( 2 x + 3 y ) ^ { 2 } - ( 15 x + y ) ( x - 5 y ) ]$$

Evaluate

$$10x\left(5y-x\right)$$

Solution Steps

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

$$9x^{2}-12xy+4y^{2}-\left(2x+3y\right)^{2}-\left(15x+y\right)\left(x-5y\right)$$

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

$$9x^{2}-12xy+4y^{2}-\left(4x^{2}+12xy+9y^{2}\right)-\left(15x+y\right)\left(x-5y\right)$$

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

$$9x^{2}-12xy+4y^{2}-4x^{2}-12xy-9y^{2}-\left(15x+y\right)\left(x-5y\right)$$

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

$$5x^{2}-12xy+4y^{2}-12xy-9y^{2}-\left(15x+y\right)\left(x-5y\right)$$

Combine -12xy and -12xy to get -24xy.

$$5x^{2}-24xy+4y^{2}-9y^{2}-\left(15x+y\right)\left(x-5y\right)$$

Combine 4y^{2} and -9y^{2} to get -5y^{2}.

$$5x^{2}-24xy-5y^{2}-\left(15x+y\right)\left(x-5y\right)$$

Use the distributive property to multiply 15x+y by x-5y and combine like terms.

$$5x^{2}-24xy-5y^{2}-\left(15x^{2}-74xy-5y^{2}\right)$$

To find the opposite of 15x^{2}-74xy-5y^{2}, find the opposite of each term.

$$5x^{2}-24xy-5y^{2}-15x^{2}+74xy+5y^{2}$$

Combine 5x^{2} and -15x^{2} to get -10x^{2}.

$$-10x^{2}-24xy-5y^{2}+74xy+5y^{2}$$

Combine -24xy and 74xy to get 50xy.

$$-10x^{2}+50xy-5y^{2}+5y^{2}$$

Combine -5y^{2} and 5y^{2} to get 0.

$$-10x^{2}+50xy$$

Expand

$$50xy-10x^{2}$$

Solution Steps

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

$$9x^{2}-12xy+4y^{2}-\left(2x+3y\right)^{2}-\left(15x+y\right)\left(x-5y\right)$$

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

$$9x^{2}-12xy+4y^{2}-\left(4x^{2}+12xy+9y^{2}\right)-\left(15x+y\right)\left(x-5y\right)$$

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

$$9x^{2}-12xy+4y^{2}-4x^{2}-12xy-9y^{2}-\left(15x+y\right)\left(x-5y\right)$$

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

$$5x^{2}-12xy+4y^{2}-12xy-9y^{2}-\left(15x+y\right)\left(x-5y\right)$$

Combine -12xy and -12xy to get -24xy.

$$5x^{2}-24xy+4y^{2}-9y^{2}-\left(15x+y\right)\left(x-5y\right)$$

Combine 4y^{2} and -9y^{2} to get -5y^{2}.

$$5x^{2}-24xy-5y^{2}-\left(15x+y\right)\left(x-5y\right)$$

Use the distributive property to multiply 15x+y by x-5y and combine like terms.

$$5x^{2}-24xy-5y^{2}-\left(15x^{2}-74xy-5y^{2}\right)$$

To find the opposite of 15x^{2}-74xy-5y^{2}, find the opposite of each term.

$$5x^{2}-24xy-5y^{2}-15x^{2}+74xy+5y^{2}$$

Combine 5x^{2} and -15x^{2} to get -10x^{2}.

$$-10x^{2}-24xy-5y^{2}+74xy+5y^{2}$$

Combine -24xy and 74xy to get 50xy.

$$-10x^{2}+50xy-5y^{2}+5y^{2}$$

Combine -5y^{2} and 5y^{2} to get 0.

$$-10x^{2}+50xy$$