Solve 4(3x-5y)^2-(4x-y)(x+y)+(2x+y)(2x-y)

$$4 ( 3 x - 5 y ) ^ { 2 } - ( 4 x - y ) ( x + y ) + ( 2 x + y ) ( 2 x - y )$$

Evaluate

$$\left(3x-4y\right)\left(12x-25y\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-5y\right)^{2}.

$$4\left(9x^{2}-30xy+25y^{2}\right)-\left(4x-y\right)\left(x+y\right)+\left(2x+y\right)\left(2x-y\right)$$

Use the distributive property to multiply 4 by 9x^{2}-30xy+25y^{2}.

$$36x^{2}-120xy+100y^{2}-\left(4x-y\right)\left(x+y\right)+\left(2x+y\right)\left(2x-y\right)$$

Use the distributive property to multiply 4x-y by x+y and combine like terms.

$$36x^{2}-120xy+100y^{2}-\left(4x^{2}+3xy-y^{2}\right)+\left(2x+y\right)\left(2x-y\right)$$

To find the opposite of 4x^{2}+3xy-y^{2}, find the opposite of each term.

$$36x^{2}-120xy+100y^{2}-4x^{2}-3xy+y^{2}+\left(2x+y\right)\left(2x-y\right)$$

Combine 36x^{2} and -4x^{2} to get 32x^{2}.

$$32x^{2}-120xy+100y^{2}-3xy+y^{2}+\left(2x+y\right)\left(2x-y\right)$$

Combine -120xy and -3xy to get -123xy.

$$32x^{2}-123xy+100y^{2}+y^{2}+\left(2x+y\right)\left(2x-y\right)$$

Combine 100y^{2} and y^{2} to get 101y^{2}.

$$32x^{2}-123xy+101y^{2}+\left(2x+y\right)\left(2x-y\right)$$

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

$$32x^{2}-123xy+101y^{2}+\left(2x\right)^{2}-y^{2}$$

Expand \left(2x\right)^{2}.

$$32x^{2}-123xy+101y^{2}+2^{2}x^{2}-y^{2}$$

Calculate 2 to the power of 2 and get 4.

$$32x^{2}-123xy+101y^{2}+4x^{2}-y^{2}$$

Combine 32x^{2} and 4x^{2} to get 36x^{2}.

$$36x^{2}-123xy+101y^{2}-y^{2}$$

Combine 101y^{2} and -y^{2} to get 100y^{2}.

$$36x^{2}-123xy+100y^{2}$$

Expand

$$36x^{2}-123xy+100y^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-5y\right)^{2}.

$$4\left(9x^{2}-30xy+25y^{2}\right)-\left(4x-y\right)\left(x+y\right)+\left(2x+y\right)\left(2x-y\right)$$

Use the distributive property to multiply 4 by 9x^{2}-30xy+25y^{2}.

$$36x^{2}-120xy+100y^{2}-\left(4x-y\right)\left(x+y\right)+\left(2x+y\right)\left(2x-y\right)$$

Use the distributive property to multiply 4x-y by x+y and combine like terms.

$$36x^{2}-120xy+100y^{2}-\left(4x^{2}+3xy-y^{2}\right)+\left(2x+y\right)\left(2x-y\right)$$

To find the opposite of 4x^{2}+3xy-y^{2}, find the opposite of each term.

$$36x^{2}-120xy+100y^{2}-4x^{2}-3xy+y^{2}+\left(2x+y\right)\left(2x-y\right)$$

Combine 36x^{2} and -4x^{2} to get 32x^{2}.

$$32x^{2}-120xy+100y^{2}-3xy+y^{2}+\left(2x+y\right)\left(2x-y\right)$$

Combine -120xy and -3xy to get -123xy.

$$32x^{2}-123xy+100y^{2}+y^{2}+\left(2x+y\right)\left(2x-y\right)$$

Combine 100y^{2} and y^{2} to get 101y^{2}.

$$32x^{2}-123xy+101y^{2}+\left(2x+y\right)\left(2x-y\right)$$

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

$$32x^{2}-123xy+101y^{2}+\left(2x\right)^{2}-y^{2}$$

Expand \left(2x\right)^{2}.

$$32x^{2}-123xy+101y^{2}+2^{2}x^{2}-y^{2}$$

Calculate 2 to the power of 2 and get 4.

$$32x^{2}-123xy+101y^{2}+4x^{2}-y^{2}$$

Combine 32x^{2} and 4x^{2} to get 36x^{2}.

$$36x^{2}-123xy+101y^{2}-y^{2}$$

Combine 101y^{2} and -y^{2} to get 100y^{2}.

$$36x^{2}-123xy+100y^{2}$$