Solve (5x+3y)(3y-5x)-(4x-y)(4y+x)

$$( 5 x + 3 y ) ( 3 y - 5 x ) - ( 4 x - y ) ( 4 y + x )$$

Evaluate

$$13y^{2}-15xy-29x^{2}$$

Solution Steps

Consider \left(5x+3y\right)\left(3y-5x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(3y\right)^{2}-\left(5x\right)^{2}-\left(4x-y\right)\left(4y+x\right)$$

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

$$3^{2}y^{2}-\left(5x\right)^{2}-\left(4x-y\right)\left(4y+x\right)$$

Calculate 3 to the power of 2 and get 9.

$$9y^{2}-\left(5x\right)^{2}-\left(4x-y\right)\left(4y+x\right)$$

Expand \left(5x\right)^{2}.

$$9y^{2}-5^{2}x^{2}-\left(4x-y\right)\left(4y+x\right)$$

Calculate 5 to the power of 2 and get 25.

$$9y^{2}-25x^{2}-\left(4x-y\right)\left(4y+x\right)$$

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

$$9y^{2}-25x^{2}-\left(16xy+4x^{2}-4y^{2}-yx\right)$$

Combine 16xy and -yx to get 15xy.

$$9y^{2}-25x^{2}-\left(15xy+4x^{2}-4y^{2}\right)$$

To find the opposite of 15xy+4x^{2}-4y^{2}, find the opposite of each term.

$$9y^{2}-25x^{2}-15xy-4x^{2}-\left(-4y^{2}\right)$$

The opposite of -4y^{2} is 4y^{2}.

$$9y^{2}-25x^{2}-15xy-4x^{2}+4y^{2}$$

Combine -25x^{2} and -4x^{2} to get -29x^{2}.

$$9y^{2}-29x^{2}-15xy+4y^{2}$$

Combine 9y^{2} and 4y^{2} to get 13y^{2}.

$$13y^{2}-29x^{2}-15xy$$

Expand

$$13y^{2}-15xy-29x^{2}$$

Solution Steps

Consider \left(5x+3y\right)\left(3y-5x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(3y\right)^{2}-\left(5x\right)^{2}-\left(4x-y\right)\left(4y+x\right)$$

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

$$3^{2}y^{2}-\left(5x\right)^{2}-\left(4x-y\right)\left(4y+x\right)$$

Calculate 3 to the power of 2 and get 9.

$$9y^{2}-\left(5x\right)^{2}-\left(4x-y\right)\left(4y+x\right)$$

Expand \left(5x\right)^{2}.

$$9y^{2}-5^{2}x^{2}-\left(4x-y\right)\left(4y+x\right)$$

Calculate 5 to the power of 2 and get 25.

$$9y^{2}-25x^{2}-\left(4x-y\right)\left(4y+x\right)$$

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

$$9y^{2}-25x^{2}-\left(16xy+4x^{2}-4y^{2}-yx\right)$$

Combine 16xy and -yx to get 15xy.

$$9y^{2}-25x^{2}-\left(15xy+4x^{2}-4y^{2}\right)$$

To find the opposite of 15xy+4x^{2}-4y^{2}, find the opposite of each term.

$$9y^{2}-25x^{2}-15xy-4x^{2}-\left(-4y^{2}\right)$$

The opposite of -4y^{2} is 4y^{2}.

$$9y^{2}-25x^{2}-15xy-4x^{2}+4y^{2}$$

Combine -25x^{2} and -4x^{2} to get -29x^{2}.

$$9y^{2}-29x^{2}-15xy+4y^{2}$$

Combine 9y^{2} and 4y^{2} to get 13y^{2}.

$$13y^{2}-29x^{2}-15xy$$