Solve (2x+y)(2x-y)(4x^{2}+y^{2})-(4x^2+y-1)^2+[(y+1)(y-1)]^2+8x^2(y-1)

$$( 2 x + y ) ( 2 x - y ) ( 4 x ^ { 2 } + y ^ { 2 } ) - ( 4 x ^ { 2 } + y - 1 ) ^ { 2 } + [ ( y + 1 ) ( y - 1 ) ] ^ { 2 } + 8 x ^ { 2 } ( y - 1 )$$

Evaluate

$$y\left(2-3y\right)$$

Solution Steps

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

$$\left(4x^{2}-y^{2}\right)\left(4x^{2}+y^{2}\right)-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

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

$$\left(4x^{2}\right)^{2}-\left(y^{2}\right)^{2}-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\left(4x^{2}\right)^{2}-y^{4}-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

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

$$4^{2}\left(x^{2}\right)^{2}-y^{4}-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4^{2}x^{4}-y^{4}-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

Calculate 4 to the power of 2 and get 16.

$$16x^{4}-y^{4}-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

Square 4x^{2}+y-1.

$$16x^{4}-y^{4}-\left(16x^{4}-8x^{2}+y^{2}+8yx^{2}-2y+1\right)+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

To find the opposite of 16x^{4}-8x^{2}+y^{2}+8yx^{2}-2y+1, find the opposite of each term.

$$16x^{4}-y^{4}-16x^{4}+8x^{2}-y^{2}-8yx^{2}+2y-1+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

Combine 16x^{4} and -16x^{4} to get 0.

$$-y^{4}+8x^{2}-y^{2}-8yx^{2}+2y-1+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

Consider \left(y+1\right)\left(y-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$-y^{4}+8x^{2}-y^{2}-8yx^{2}+2y-1+\left(y^{2}-1\right)^{2}+8x^{2}\left(y-1\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y^{2}-1\right)^{2}.

$$-y^{4}+8x^{2}-y^{2}-8yx^{2}+2y-1+\left(y^{2}\right)^{2}-2y^{2}+1+8x^{2}\left(y-1\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$-y^{4}+8x^{2}-y^{2}-8yx^{2}+2y-1+y^{4}-2y^{2}+1+8x^{2}\left(y-1\right)$$

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

$$8x^{2}-y^{2}-8yx^{2}+2y-1-2y^{2}+1+8x^{2}\left(y-1\right)$$

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

$$8x^{2}-3y^{2}-8yx^{2}+2y-1+1+8x^{2}\left(y-1\right)$$

Add -1 and 1 to get 0.

$$8x^{2}-3y^{2}-8yx^{2}+2y+8x^{2}\left(y-1\right)$$

Use the distributive property to multiply 8x^{2} by y-1.

$$8x^{2}-3y^{2}-8yx^{2}+2y+8x^{2}y-8x^{2}$$

Combine -8yx^{2} and 8x^{2}y to get 0.

$$8x^{2}-3y^{2}+2y-8x^{2}$$

Combine 8x^{2} and -8x^{2} to get 0.

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

Expand

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

Solution Steps

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

$$\left(4x^{2}-y^{2}\right)\left(4x^{2}+y^{2}\right)-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

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

$$\left(4x^{2}\right)^{2}-\left(y^{2}\right)^{2}-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\left(4x^{2}\right)^{2}-y^{4}-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

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

$$4^{2}\left(x^{2}\right)^{2}-y^{4}-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4^{2}x^{4}-y^{4}-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

Calculate 4 to the power of 2 and get 16.

$$16x^{4}-y^{4}-\left(4x^{2}+y-1\right)^{2}+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

Square 4x^{2}+y-1.

$$16x^{4}-y^{4}-\left(16x^{4}-8x^{2}+y^{2}+8yx^{2}-2y+1\right)+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

To find the opposite of 16x^{4}-8x^{2}+y^{2}+8yx^{2}-2y+1, find the opposite of each term.

$$16x^{4}-y^{4}-16x^{4}+8x^{2}-y^{2}-8yx^{2}+2y-1+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

Combine 16x^{4} and -16x^{4} to get 0.

$$-y^{4}+8x^{2}-y^{2}-8yx^{2}+2y-1+\left(\left(y+1\right)\left(y-1\right)\right)^{2}+8x^{2}\left(y-1\right)$$

Consider \left(y+1\right)\left(y-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$-y^{4}+8x^{2}-y^{2}-8yx^{2}+2y-1+\left(y^{2}-1\right)^{2}+8x^{2}\left(y-1\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y^{2}-1\right)^{2}.

$$-y^{4}+8x^{2}-y^{2}-8yx^{2}+2y-1+\left(y^{2}\right)^{2}-2y^{2}+1+8x^{2}\left(y-1\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$-y^{4}+8x^{2}-y^{2}-8yx^{2}+2y-1+y^{4}-2y^{2}+1+8x^{2}\left(y-1\right)$$

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

$$8x^{2}-y^{2}-8yx^{2}+2y-1-2y^{2}+1+8x^{2}\left(y-1\right)$$

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

$$8x^{2}-3y^{2}-8yx^{2}+2y-1+1+8x^{2}\left(y-1\right)$$

Add -1 and 1 to get 0.

$$8x^{2}-3y^{2}-8yx^{2}+2y+8x^{2}\left(y-1\right)$$

Use the distributive property to multiply 8x^{2} by y-1.

$$8x^{2}-3y^{2}-8yx^{2}+2y+8x^{2}y-8x^{2}$$

Combine -8yx^{2} and 8x^{2}y to get 0.

$$8x^{2}-3y^{2}+2y-8x^{2}$$

Combine 8x^{2} and -8x^{2} to get 0.

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