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

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

Evaluate

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

Solution Steps

Consider \left(4x-3y\right)\left(4x+3y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-\left(3y\right)^{2}-4\left(2x-y\right)\left(2x+y\right)+5\left(x^{2}+2y^{2}\right)$$

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

$$16x^{2}-3^{2}y^{2}-4\left(2x-y\right)\left(2x+y\right)+5\left(x^{2}+2y^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$16x^{2}-9y^{2}-4\left(2x-y\right)\left(2x+y\right)+5\left(x^{2}+2y^{2}\right)$$

Use the distributive property to multiply 5 by x^{2}+2y^{2}.

$$16x^{2}-9y^{2}-4\left(2x-y\right)\left(2x+y\right)+5x^{2}+10y^{2}$$

Use the distributive property to multiply -4 by 2x-y.

$$16x^{2}-9y^{2}+\left(-8x+4y\right)\left(2x+y\right)+5x^{2}+10y^{2}$$

Use the distributive property to multiply -8x+4y by 2x+y and combine like terms.

$$16x^{2}-9y^{2}-16x^{2}+4y^{2}+5x^{2}+10y^{2}$$

Combine 16x^{2} and -16x^{2} to get 0.

$$-9y^{2}+4y^{2}+5x^{2}+10y^{2}$$

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

$$-5y^{2}+5x^{2}+10y^{2}$$

Combine -5y^{2} and 10y^{2} to get 5y^{2}.

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

Expand

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

Solution Steps

Consider \left(4x-3y\right)\left(4x+3y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-\left(3y\right)^{2}-4\left(2x-y\right)\left(2x+y\right)+5\left(x^{2}+2y^{2}\right)$$

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

$$16x^{2}-3^{2}y^{2}-4\left(2x-y\right)\left(2x+y\right)+5\left(x^{2}+2y^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$16x^{2}-9y^{2}-4\left(2x-y\right)\left(2x+y\right)+5\left(x^{2}+2y^{2}\right)$$

Use the distributive property to multiply 5 by x^{2}+2y^{2}.

$$16x^{2}-9y^{2}-4\left(2x-y\right)\left(2x+y\right)+5x^{2}+10y^{2}$$

Use the distributive property to multiply -4 by 2x-y.

$$16x^{2}-9y^{2}+\left(-8x+4y\right)\left(2x+y\right)+5x^{2}+10y^{2}$$

Use the distributive property to multiply -8x+4y by 2x+y and combine like terms.

$$16x^{2}-9y^{2}-16x^{2}+4y^{2}+5x^{2}+10y^{2}$$

Combine 16x^{2} and -16x^{2} to get 0.

$$-9y^{2}+4y^{2}+5x^{2}+10y^{2}$$

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

$$-5y^{2}+5x^{2}+10y^{2}$$

Combine -5y^{2} and 10y^{2} to get 5y^{2}.

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