Answer
$= \displaystyle \frac{7}{9},\qquad x\neq-1$
Work Step by Step
Division with $\displaystyle \frac{a}{b}$ equals multiplication with $\displaystyle \frac{b}{a}.$
(neither a or b can be 0)
$\displaystyle \frac{x+1}{3}\div\frac{3x+3}{7}==\frac{x+1}{3}\cdot\frac{7}{3x+3}$
... Factor what we can
$=\displaystyle \frac{(x+1)}{3}\cdot\frac{7}{3(x+1)}$
...exclude the values that yield 0 in the denominator:
$x\neq-1$
... cancel common factors
$= \displaystyle \frac{7}{9},\qquad x\neq-1$
