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

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

Evaluate

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

Solution Steps

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

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

Use the distributive property to multiply 4 by x^{2}+2xy+y^{2}.

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

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

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

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

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

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

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

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

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

Combine 4y^{2} and 4y^{2} to get 8y^{2}.

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

Combine 8xy and -2xy to get 6xy.

$$6xy+8y^{2}+x^{2}+y^{2}$$

Combine 8y^{2} and y^{2} to get 9y^{2}.

$$6xy+9y^{2}+x^{2}$$

Expand

$$x^{2}+6xy+9y^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 4 by x^{2}+2xy+y^{2}.

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

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

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

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

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

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

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

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

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

Combine 4y^{2} and 4y^{2} to get 8y^{2}.

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

Combine 8xy and -2xy to get 6xy.

$$6xy+8y^{2}+x^{2}+y^{2}$$

Combine 8y^{2} and y^{2} to get 9y^{2}.

$$6xy+9y^{2}+x^{2}$$