<s>[INST] The distributive law $4\left (x+y \right)=4x+4y $ because $4\left (x+y \right) =\left (x+y \right) +\left (x+y \right) +\left (x+y \right) +\left (x+y \right)$ , but why is $\left (x+y \right) \left (x+y \right) =xx+xy+yx+yy$?
 [/INST] Do it in steps.  
You accept that $M(x+y) = Mx + My$
So replace $M$ with $(x+y)$ and you get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y$
Now distribute a second time:  Replace $x$ with $A$ and $y$ with $B$ to get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y=$
$(x+y)A + (x+y)B=$
$xA + yA + xB + yB =$
$xx + yx + xy + yy =$
$x^2 + 2xy + y^2$.
Of you you don't have to, and you shouldn't, do all that replacement.  You should do it directly. 
$(x+y)(x+y)=$ we treat one of the $(x+y)$ as a single thing and distribute across the other $x + y$.
$(x+y)(x+y) = (x+y)x + (x+y)y=$.
Now we have to sums to distribute: $(x+y)x = xx + yx$  and $(x + y)y = xy + yy$.  So puting them together:
$(x+y)(x+y) = (x+y)x + (x+y)y=xx + yx + xy + yy=$.
And then some clean up:
$=x^2 + 2xy + y^2$
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