Can somebody help me prove that $x^{\underline{k+1}}+kx^{\underline{k}} = xx^{\underline{k}}$?

Question

I need to prove that $x^{\underline{k+1}}+kx^{\underline{k}} = xx^{\underline{k}}$, where $x^{\underline{k}}$ is $x$ to the falling $k$ factorial, and I have no idea where to start. A nudge in the right direction would be much appreciated. Thanks!

Answer 1

$$ x^{\underline{k}}=x(x-1)\cdots(x-k+1) $$

$$ x^{\underline{k+1}}=x(x-1)\cdots(x-k+1)(x-(k+1)+1)=x(x-1)\cdots(x-k+1)(x-k) $$

$$ x^{\underline{k+1}}=x^{\underline{k}}(x-k) $$

Do you see how to complete the problem from here?

Answer 2

$$x^{\underline{k+1}} = x^{\underline{k}}(x-k).$$