If a polynomial P is an odd function, then P(-x)=-P(x).

So if you add a constant, say m, then let Q(x)=P(x)+m.

The thing is now to prove that Q is not odd.

Q(-x)=P(-x)+m=-P(x)+m -Q(x)=-(P(x)+m)

Do you quite understand ?

Graphically, adding a constant means to level up (or down) the curve. Will it still be symmetric wrt the center of the graph ?