Solve (3x+2y)(3x-2y)-(x+2y)(5x-2y)]div8x

$$( 3 x + 2 y ) ( 3 x - 2 y ) - ( x + 2 y ) ( 5 x - 2 y ) ] \div 8 x$$

Evaluate

$$4x\left(x-2y\right)$$

Solution Steps

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}.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Apply the distributive property by multiplying each term of x+2y by each term of 5x-2y.

$$9x^{2}-4y^{2}-\left(5x^{2}-2xy+10yx-4y^{2}\right)$$

Combine -2xy and 10yx to get 8xy.

$$9x^{2}-4y^{2}-\left(5x^{2}+8xy-4y^{2}\right)$$

To find the opposite of 5x^{2}+8xy-4y^{2}, find the opposite of each term.

$$9x^{2}-4y^{2}-5x^{2}-8xy-\left(-4y^{2}\right)$$

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

$$9x^{2}-4y^{2}-5x^{2}-8xy+4y^{2}$$

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

$$4x^{2}-4y^{2}-8xy+4y^{2}$$

Combine -4y^{2} and 4y^{2} to get 0.

$$4x^{2}-8xy$$

Expand

$$4x^{2}-8xy$$

Solution Steps

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}.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

Apply the distributive property by multiplying each term of x+2y by each term of 5x-2y.

$$9x^{2}-4y^{2}-\left(5x^{2}-2xy+10yx-4y^{2}\right)$$

Combine -2xy and 10yx to get 8xy.

$$9x^{2}-4y^{2}-\left(5x^{2}+8xy-4y^{2}\right)$$

To find the opposite of 5x^{2}+8xy-4y^{2}, find the opposite of each term.

$$9x^{2}-4y^{2}-5x^{2}-8xy-\left(-4y^{2}\right)$$

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

$$9x^{2}-4y^{2}-5x^{2}-8xy+4y^{2}$$

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

$$4x^{2}-4y^{2}-8xy+4y^{2}$$

Combine -4y^{2} and 4y^{2} to get 0.

$$4x^{2}-8xy$$