Expand and Simplify: Solving (x+y)²-(x-y)²+(x-y)(x+y)

$(x+y)^2-(x-y)^2+(x-y)(x+y)=\text{?}$

To solve this problem, we'll follow these steps:

• Step 1: Expand $(x+y)^2$
• Step 2: Expand $(x-y)^2$
• Step 3: Rearrange and simplify the entire expression

Now, let's work through each step:
Step 1: We expand $(x+y)^2$ using the formula for the square of a sum:
$(x+y)^2 = x^2 + 2xy + y^2$.

Step 2: We expand $(x-y)^2$ using the formula for the square of a difference:
$(x-y)^2 = x^2 - 2xy + y^2$.

Step 3: Substitute these expansions back into the original expression: $(x+y)^2-(x-y)^2+(x-y)(x+y)$ becomes: $(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) + (x-y)(x+y).$

First, simplify $(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)$:
– $x^2 + 2xy + y^2 - x^2 + 2xy - y^2 = 4xy.$

Next, consider $(x-y)(x+y)$:
By using the identity for difference of squares: $(x-y)(x+y) = x^2 - y^2$.

Thus, combining our results gives:
$4xy + x^2 - y^2 = x^2 + 4xy - y^2.$

Therefore, the solution to the problem is $x^2 + 4xy - y^2$.