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

$$(3x-y)(x+2y)-(x+y)(x-y)$$

Evaluate

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

Solution Steps

Apply the distributive property by multiplying each term of 3x-y by each term of x+2y.

$$3x^{2}+6xy-yx-2y^{2}-\left(x+y\right)\left(x-y\right)$$

Combine 6xy and -yx to get 5xy.

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

Consider \left(x+y\right)\left(x-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}.

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

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

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

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

$$3x^{2}+5xy-2y^{2}-x^{2}+y^{2}$$

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

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

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

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

Expand

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

Solution Steps

Apply the distributive property by multiplying each term of 3x-y by each term of x+2y.

$$3x^{2}+6xy-yx-2y^{2}-\left(x+y\right)\left(x-y\right)$$

Combine 6xy and -yx to get 5xy.

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

Consider \left(x+y\right)\left(x-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}.

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

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

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

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

$$3x^{2}+5xy-2y^{2}-x^{2}+y^{2}$$

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

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

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

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