MHB Solve Factoring Problem: (x+y)^2+2(X+Y)+1

[caligari]

The problem is
(x+y)^2+2(X+Y)+1

and the answer is supposed to be (X+1+Y)^2

I looked it up on this calculator but the second step makes no sense. First I do not know how the +2 disappears and where the extra (X+Y) come from. Also it's explanation when you click on the black box doesn't make sense. It says "For a quadratic equation of the form ax2+bx+c find u,v such that: u(v)=a(c) and u+v=c.
What are u and c and also in the next step it shows it as this
((x+y)+1))((x+y)^2+(x+y)) If the hint tells me to put it in ax2+bx+c then why are there only two terms in this. THis step makes no sense.
Here is the link to the calculator and problem

https://www.symbolab.com/solver/abs...right)^{2}+2\left(x+y\right)+1/?origin=button

 

[MarkFL]

Hello and welcome to MHB, caligari! :D

We are given to factor:

$$(x+y)^2+2(x+y)+1$$

Now, suppose we let $u=x+y$, and then we have:

$$u^2+2u+1$$

You should recognize this as the square of $u+1$, hence:

$$u^2+2u+1=(u+1)^2$$

And so, back-substituting for $u$, we obtain:

$$(x+y)^2+2(x+y)+1=(x+y+1)^2$$

 

[topsquark]

The problem is
(x+y)^2+2(X+Y)+1

Just a quick note:

Mathematics is "case sensitive." X and x are not the same variable.

-Dan