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

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

Evaluate

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

Solution Steps

Consider \left(x+3y\right)\left(x-3y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(3y\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

Use the distributive property to multiply -2 by x+2y.

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

Apply the distributive property by multiplying each term of -2x-4y by each term of x-y.

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

Combine 2xy and -4yx to get -2xy.

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

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

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

Combine -9y^{2} and 4y^{2} to get -5y^{2}.

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

Expand

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

Solution Steps

Consider \left(x+3y\right)\left(x-3y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(3y\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

Use the distributive property to multiply -2 by x+2y.

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

Apply the distributive property by multiplying each term of -2x-4y by each term of x-y.

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

Combine 2xy and -4yx to get -2xy.

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

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

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

Combine -9y^{2} and 4y^{2} to get -5y^{2}.

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