Solve (5x+y)(y-5x)-(3x-y)^2+(30x-6y)(x+1)

$$( 5 x + y ) ( y - 5 x ) - ( 3 x - y ) ^ { 2 } + ( 30 x - 6 y ) ( x + 1 )$$

Evaluate

$$-4x^{2}+30x-6y$$

Solution Steps

Consider \left(5x+y\right)\left(y-5x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$y^{2}-\left(5x\right)^{2}-\left(3x-y\right)^{2}+\left(30x-6y\right)\left(x+1\right)$$

Expand \left(5x\right)^{2}.

$$y^{2}-5^{2}x^{2}-\left(3x-y\right)^{2}+\left(30x-6y\right)\left(x+1\right)$$

Calculate 5 to the power of 2 and get 25.

$$y^{2}-25x^{2}-\left(3x-y\right)^{2}+\left(30x-6y\right)\left(x+1\right)$$

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

$$y^{2}-25x^{2}-\left(9x^{2}-6xy+y^{2}\right)+\left(30x-6y\right)\left(x+1\right)$$

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

$$y^{2}-25x^{2}-9x^{2}+6xy-y^{2}+\left(30x-6y\right)\left(x+1\right)$$

Combine -25x^{2} and -9x^{2} to get -34x^{2}.

$$y^{2}-34x^{2}+6xy-y^{2}+\left(30x-6y\right)\left(x+1\right)$$

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

$$-34x^{2}+6xy+\left(30x-6y\right)\left(x+1\right)$$

Use the distributive property to multiply 30x-6y by x+1.

$$-34x^{2}+6xy+30x^{2}+30x-6yx-6y$$

Combine -34x^{2} and 30x^{2} to get -4x^{2}.

$$-4x^{2}+6xy+30x-6yx-6y$$

Combine 6xy and -6yx to get 0.

$$-4x^{2}+30x-6y$$

Expand

$$-4x^{2}+30x-6y$$

Solution Steps

Consider \left(5x+y\right)\left(y-5x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$y^{2}-\left(5x\right)^{2}-\left(3x-y\right)^{2}+\left(30x-6y\right)\left(x+1\right)$$

Expand \left(5x\right)^{2}.

$$y^{2}-5^{2}x^{2}-\left(3x-y\right)^{2}+\left(30x-6y\right)\left(x+1\right)$$

Calculate 5 to the power of 2 and get 25.

$$y^{2}-25x^{2}-\left(3x-y\right)^{2}+\left(30x-6y\right)\left(x+1\right)$$

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

$$y^{2}-25x^{2}-\left(9x^{2}-6xy+y^{2}\right)+\left(30x-6y\right)\left(x+1\right)$$

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

$$y^{2}-25x^{2}-9x^{2}+6xy-y^{2}+\left(30x-6y\right)\left(x+1\right)$$

Combine -25x^{2} and -9x^{2} to get -34x^{2}.

$$y^{2}-34x^{2}+6xy-y^{2}+\left(30x-6y\right)\left(x+1\right)$$

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

$$-34x^{2}+6xy+\left(30x-6y\right)\left(x+1\right)$$

Use the distributive property to multiply 30x-6y by x+1.

$$-34x^{2}+6xy+30x^{2}+30x-6yx-6y$$

Combine -34x^{2} and 30x^{2} to get -4x^{2}.

$$-4x^{2}+6xy+30x-6yx-6y$$

Combine 6xy and -6yx to get 0.

$$-4x^{2}+30x-6y$$