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

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

Evaluate

$$x\left(3x-y\right)$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}y+xy^{2}-y^{3}}{y}.

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

Cancel out y in both numerator and denominator.

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

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

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

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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}-y^{2}-xy+y^{2}$$

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

$$3x^{2}-xy$$

Expand

$$3x^{2}-xy$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}y+xy^{2}-y^{3}}{y}.

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

Cancel out y in both numerator and denominator.

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

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

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

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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}-y^{2}-xy+y^{2}$$

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

$$3x^{2}-xy$$