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

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

Evaluate

$$10x^{2}+xy-7y^{2}$$

Solution Steps

Consider \left(y+3x\right)\left(3x-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}.

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

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

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

Calculate 3 to the power of 2 and get 9.

$$9x^{2}-y^{2}-\left(2y-x\right)\left(3y+x\right)$$

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

$$9x^{2}-y^{2}-\left(6y^{2}+2yx-3xy-x^{2}\right)$$

Combine 2yx and -3xy to get -yx.

$$9x^{2}-y^{2}-\left(6y^{2}-yx-x^{2}\right)$$

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

$$9x^{2}-y^{2}-6y^{2}-\left(-yx\right)-\left(-x^{2}\right)$$

The opposite of -yx is yx.

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

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

$$9x^{2}-y^{2}-6y^{2}+yx+x^{2}$$

Combine -y^{2} and -6y^{2} to get -7y^{2}.

$$9x^{2}-7y^{2}+yx+x^{2}$$

Combine 9x^{2} and x^{2} to get 10x^{2}.

$$10x^{2}-7y^{2}+yx$$

Expand

$$10x^{2}+xy-7y^{2}$$

Solution Steps

Consider \left(y+3x\right)\left(3x-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}.

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

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

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

Calculate 3 to the power of 2 and get 9.

$$9x^{2}-y^{2}-\left(2y-x\right)\left(3y+x\right)$$

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

$$9x^{2}-y^{2}-\left(6y^{2}+2yx-3xy-x^{2}\right)$$

Combine 2yx and -3xy to get -yx.

$$9x^{2}-y^{2}-\left(6y^{2}-yx-x^{2}\right)$$

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

$$9x^{2}-y^{2}-6y^{2}-\left(-yx\right)-\left(-x^{2}\right)$$

The opposite of -yx is yx.

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

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

$$9x^{2}-y^{2}-6y^{2}+yx+x^{2}$$

Combine -y^{2} and -6y^{2} to get -7y^{2}.

$$9x^{2}-7y^{2}+yx+x^{2}$$

Combine 9x^{2} and x^{2} to get 10x^{2}.

$$10x^{2}-7y^{2}+yx$$