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

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

Evaluate

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

Solution Steps

Consider \left(2x-y\right)\left(2x+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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

To find the opposite of 9x^{2}-4y^{2}, find the opposite of each term.

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

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

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

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

$$-5x^{2}-y^{2}+4y^{2}$$

Combine -y^{2} and 4y^{2} to get 3y^{2}.

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

Expand

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

Solution Steps

Consider \left(2x-y\right)\left(2x+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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

To find the opposite of 9x^{2}-4y^{2}, find the opposite of each term.

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

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

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

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

$$-5x^{2}-y^{2}+4y^{2}$$

Combine -y^{2} and 4y^{2} to get 3y^{2}.

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