Solve (x+y)^{2}(x-y)^{2}+2y^{2}(x^{2}-y^{2})-(x^2-y^2)(x^2+y^2)

$$( x + y ) ^ { 2 } ( x - y ) ^ { 2 } + 2 y ^ { 2 } ( x ^ { 2 } - y ^ { 2 } ) - ( x ^ { 2 } - y ^ { 2 } ) ( x ^ { 2 } + y ^ { 2 } )$$

Evaluate

$$0$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$-y^{4}+y^{4}$$

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

$$0$$

Factor

$$0$$