Answer
$ \frac{-(x-y)}{(x+y)(b+c)}$.
Work Step by Step
The given expression is
$\Rightarrow \frac{ax-ay+3x-3y}{x^3+y^3}\div \frac{ab+3b+ac+3c}{xy-x^2-y^2}$
Invert the divisor and multiply.
$\Rightarrow \frac{ax-ay+3x-3y}{x^3+y^3}\cdot \frac{xy-x^2-y^2}{ab+3b+ac+3c}$
Factor each numerator and denominator as shown below.
$\Rightarrow ax-ay+3x-3y$
Group terms.
$\Rightarrow (ax-ay)+(3x-3y)$
Factor each group.
$\Rightarrow a(x-y)+3(x-y)$
Factor out $(x-y)$.
$\Rightarrow (x-y)(a+3)$
$\Rightarrow x^3+y^3$
Use the formula $a^3+b^3=(a+b)(a^2-ab+b^2)$.
$\Rightarrow (x+y)(x^2-xy+y^2)$
$\Rightarrow xy-x^2-y^2$
Factor out $-1$ from all terms.
$\Rightarrow -1(x^2-xy+y^2)$
$\Rightarrow ab+3b+ac+3c$
Group terms.
$\Rightarrow (ab+3b)+(ac+3c)$
Factor each group.
$\Rightarrow b(a+3)+c(a+3)$
Factor out $(a+3)$.
$\Rightarrow (a+3)(b+c)$
Substitute all the factors into the given expression.
$\Rightarrow \frac{(x-y)(a+3)}{(x+y)(x^2-xy+y^2)}\cdot \frac{-1(x^2-xy+y^2)}{(a+3)(b+c)}$
Cancel common terms.
$\Rightarrow \frac{-(x-y)}{(x+y)(b+c)}$.
