Why is it so? (Check out the image first)
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Claim:
$$(x+a)\sum_{k=0}^{2n}(-1)^k x^k a^{2n-k} = x^{2n+1}+a^{2n+1}$$
Proof:
$$(x+a)\sum_{k=0}^{2n}(-1)^k x^k a^{2n-k} = (x+a)(x^{2n}-ax^{2n-1}+a^2x^{2n-2}\pm \cdots - a^{2n-1}x+a^{2n})$$
Now, let's expand and line up terms (the top line is the summation multiplied by $x$ while the bottom line is the summation multiplied by $a$):
$$\begin{matrix} & x^{2n+1} & -ax^{2n} & + a^2x^{2n-1} & \pm \cdots & -a^{2n-1}x^2 & +a^{2n}x & \\ + & & ax^{2n} & -a^2x^{2n-1} & \pm \cdots & +a^{2n-1}x^2 & -a^{2n}x & +a^{2n+1}\end{matrix}$$
Note that the central terms all cancel out, and all that is left is $x^{2n+1}+a^{2n+1}$.
Can you do something similar for $x^{2n}-a^{2n}$?