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

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

Evaluate

$$-5x^{2}+6xy-8y^{2}$$

Solution Steps

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

$$y^{2}-\left(2x\right)^{2}-\left(x-3y\right)^{2}$$

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

$$y^{2}-2^{2}x^{2}-\left(x-3y\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

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

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

$$y^{2}-4x^{2}-\left(x^{2}-6xy+9y^{2}\right)$$

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

$$y^{2}-4x^{2}-x^{2}+6xy-9y^{2}$$

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

$$y^{2}-5x^{2}+6xy-9y^{2}$$

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

$$-8y^{2}-5x^{2}+6xy$$

Expand

$$-5x^{2}+6xy-8y^{2}$$

Solution Steps

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

$$y^{2}-\left(2x\right)^{2}-\left(x-3y\right)^{2}$$

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

$$y^{2}-2^{2}x^{2}-\left(x-3y\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

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

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

$$y^{2}-4x^{2}-\left(x^{2}-6xy+9y^{2}\right)$$

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

$$y^{2}-4x^{2}-x^{2}+6xy-9y^{2}$$

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

$$y^{2}-5x^{2}+6xy-9y^{2}$$

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

$$-8y^{2}-5x^{2}+6xy$$