I have reduced a knotty research problem to the following reasonable looking form:

Given any two integers $a$ and $b$, show that there are natural numbers $x_1,x_2,x_3$ and an integer $n$, where $3n < x_1+x_2+x_3$, satisfying:

$x_1x_2x_3=n^3+an+b,$ and

$x_1x_2+x_1x_3+x_2x_3=a+3n^2.$

I am not expecting a solution to this (although that would of course be the ideal outcome)! However, I don't really know where to start. How might one go about solving something like this? Are there any tried and tested methods I should know about?

And finally, given the unsolvability of Hilbert's tenth problem, is it possible that there is no way to know whether or not this is true?

(edit: equations corrected. Sorry for time-wasting!)