Answer
$=\displaystyle \frac{2(x+3)}{3},\qquad x\neq 3, -3$
Work Step by Step
Factor what we can.
$\displaystyle \frac{4x^{2}+10}{x-3}\div\frac{6x^{2}+15}{x^{2}-9}=\frac{2(2x^{2}+5)}{(x-3)}\div\frac{3(2x^{2}+5)}{(x-3)(x+3)}$
Division with $\displaystyle \frac{a}{b}$ equals multiplication with $\displaystyle \frac{b}{a}.$
(neither a or b can be 0,
because of the denominator we exclude $-3$ and $3$,
$2x^{2}+5$ can never be 0 so there is nothing to exclude there)
$=\displaystyle \frac{2(2x^{2}+5)}{(x-3)}\cdot\frac{(x-3)(x+3)}{3(2x^{2}+5)}$
...exclude any additional values that yield 0 in the denominator:
$x\neq 3, -3$
... cancel common factors
$=\displaystyle \frac{2(x+3)}{3},\qquad x\neq 3, -3$
