Conversion algorithm needed.

Question

I just started an online course in math, and have a problem with an exercise. I don't know what this type of math is called in English, so please forgive me for the bad title (and tags).

The question:

I know that

$$x^2 + y^2 + 2xy = (x+y)^2.$$

But I need an explanation for how to reach the result. If someone would explain it to me like I am $5$ years old, I would grateful.

Edited:

Btw. I do know how to do it the way around: $$(x+y)^2 = (x+y)*(x+y) =$$ (Now comes the conversion I don't know how to do it the other way around) $$(x*x)+(x*y)+(y*x)+(y*y) = ....$$

Answer 1

$$\begin{align}(x + y) ^2 &= (x + y) \ (x + y) \\ &= x \ (x + y)+ y \ (x + y) \\ &= \cdots \end{align}$$

Answer 2

Before showing you how to arrive at your solution, let's start with what you already know:

$$\begin{align}(x + y) ^2 &= (x + y) \ (x + y) \\ &= x \ (x + y)+ y \ (x + y) \\ &= x^2 + xy + yx + y^2 \\ &= x^2 + y^2 + xy + yx\\ &= x^2 + y^2 + 2xy \end{align}$$

The process for showing the inverse will be to walk the steps backwards.

$$\begin{align}x^2 + y^2 + 2xy &= x^2+y^2 + xy + xy \\ &= x^2 + xy + yx + y^2 \\ &= x \ (x+y) + y \ (x + y)\\ &= (x + y)(x + y) \\ &= (x + y)^2 \end{align}$$