How do you simplify (4x^3 - 3x - 7) - (3x^3 + 5x + 3)?

Regroup terms with common powers of $x$ and do the arithmetic normally.
$\left(\textcolor{b l u e}{4 {x}^{3} - 3 x - 7}\right) - \left(\textcolor{red}{3 {x}^{3} + 5 x + 3}\right)$
$= \left(\textcolor{b l u e}{+ 4 {x}^{3}} - \textcolor{red}{3 {x}^{3}}\right) + \left(\textcolor{b l u e}{- 3 x} - \textcolor{red}{5 x}\right) + \left(\textcolor{b l u e}{- 7} - \textcolor{red}{3}\right)$
$= {x}^{3} - 8 x - 10$