Square of Sum/Algebraic Proof 1

Theorem

 * $\forall x, y \in \R: \left({x + y}\right)^2 = x^2 + 2 x y + y^2$

Proof
Follows from the distribution of multiplication over addition:

More succinctly, it follows directly from the Binomial Theorem:
 * $\displaystyle \forall n \in \Z_{>0}: \left({x+y}\right)^n = \sum_{k \mathop = 0}^n \binom n k x^{n-k}y^k$

putting $n = 2$.