Even and odd functions

Q: Show that every polynomial is the sum of an even and an odd function.
Here is what I've done, but it feels more like I have just stated stuff rather than shown it:
https://ibb.co/ZMPT5hf

How can I show this result?

Possibly show that the odd power terms satisfy the definition of an odd function and similarly for even power terms and therefore the sums are also odd/even, but that may be overkill? You've got the right idea.

Edit. The more general result is if g(x) is even and h(x) is odd then
f(x) = g(x) + h(x)
Where
g(x) = (f(x)+f(-x))/2
h(x) = (f(x)-f(-x))/2
For any function f(x).

I've got nothing to add... but please don't write arbitrary polynomials like that.

$\ f(x)= a_{0} + a_{1} x + … + a_{n-1} x^{n-1} + a_{n} x^{n}$

Is just so much better looking.