Multiply by the reciprocal of the divisor. To simplify, first factor the numerator and denominator. Then regroup common factors in the same fraction and reduce to 1.

$\displaystyle\frac{{{6}{{x}}^{{3}}}}{{25}}\div\frac{{{9}{{x}}^{{4}}}}{{20}}=\frac{{{6}{{x}}^{{3}}}}{{25}}\cdot\frac{{20}}{{{9}{{x}}^{{4}}}}=\frac{{{2}\cdot{3}\cdot{{x}}^{{3}}}}{{{5}\cdot{5}}}\cdot\frac{{{2}\cdot{2}\cdot{5}}}{{{3}\cdot{3}\cdot{{x}}^{{3}}\cdot{x}}}\usepackage{cmbright}\Huge\$

$\displaystyle=\frac{{{3}\cdot{5}\cdot{{x}}^{{3}}}}{{{3}\cdot{5}\cdot{{x}}^{{3}}}}\cdot\frac{{{2}\cdot{2}\cdot{2}}}{{{3}\cdot{5}\cdot{x}}}={1}\cdot\frac{{8}}{{{15}{x}}}=\frac{{8}}{{{15}{x}}}\usepackage{cmbright}\Huge\$