Solve (2x-3y)(3y+2x)-(5y-4x)(4x+5y)

$$( 2 x - 3 y ) ( 3 y + 2 x ) - ( 5 y - 4 x ) ( 4 x + 5 y )$$

Evaluate

$$20x^{2}-34y^{2}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

Consider \left(5y-4x\right)\left(4x+5y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 5 to the power of 2 and get 25.

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

Expand \left(4x\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

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

To find the opposite of 25y^{2}-16x^{2}, find the opposite of each term.

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

The opposite of -16x^{2} is 16x^{2}.

$$4x^{2}-9y^{2}-25y^{2}+16x^{2}$$

Combine -9y^{2} and -25y^{2} to get -34y^{2}.

$$4x^{2}-34y^{2}+16x^{2}$$

Combine 4x^{2} and 16x^{2} to get 20x^{2}.

$$20x^{2}-34y^{2}$$

Expand

$$20x^{2}-34y^{2}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

Consider \left(5y-4x\right)\left(4x+5y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 5 to the power of 2 and get 25.

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

Expand \left(4x\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

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

To find the opposite of 25y^{2}-16x^{2}, find the opposite of each term.

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

The opposite of -16x^{2} is 16x^{2}.

$$4x^{2}-9y^{2}-25y^{2}+16x^{2}$$

Combine -9y^{2} and -25y^{2} to get -34y^{2}.

$$4x^{2}-34y^{2}+16x^{2}$$

Combine 4x^{2} and 16x^{2} to get 20x^{2}.

$$20x^{2}-34y^{2}$$