Well, here we go, let's test that against the smallest nonfield von Neumann regular ring $R=F_2\times F_2$ where $F_2$ is the field of two elements.
Let $x=a=(1,0)$. You get
$1+x=(0,1)$ which are not equal.
From what you've written it really looks like you've unclearly stated your hypotheses, but in this example $axa=a$ and $xax=x$, so it covers pretty much every interpretation of what you wrote.
So, something is wrong with the statement.
If, for example, you meant that $a$ and $x$ have to be related by $a=axa$, then it would be trivial to show
which is as close as I could get to the suggested equality.
And if you meant for $xax=x$, that turns into
$ax(1+x)=ax+x=a(1+x)$ also, of course.