How do you factor -24x^3 + 2x + 47x^2?

Given $- 24 {x}^{3} + 2 x + 47 {x}^{2}$
Extract the obvious common term $\left(- x\right)$ and re-arrange in normal sequence:
$= \left(- x\right) \left(24 {x}^{2} - 47 x - 2\right)$
Recognize that $\left(- 47\right) = \left(\textcolor{red}{24} \times \textcolor{b l u e}{\left(- 2\right)}\right) + \left(\textcolor{red}{1} \times \textcolor{b l u e}{1}\right)$
$= \left(- x\right) \left(\textcolor{red}{24 x + 1}\right) \left(\textcolor{b l u e}{1 x - 2}\right)$