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

$$( 2 x - y ) ( y + 2 x ) - ( 2 y + x ) ( 2 y - x )$$

Evaluate

$$5\left(x^{2}-y^{2}\right)$$

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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

The opposite of -x^{2} is x^{2}.

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

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

$$4x^{2}-5y^{2}+x^{2}$$

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

$$5x^{2}-5y^{2}$$

Expand

$$5x^{2}-5y^{2}$$

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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

The opposite of -x^{2} is x^{2}.

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

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

$$4x^{2}-5y^{2}+x^{2}$$

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

$$5x^{2}-5y^{2}$$