dividing rational expressions
Dividing rational expressions reuses the oldest trick in the fraction book: to divide by a fraction, flip it and multiply. Where you once wrote 2/3 ÷ 4/5 = 2/3 · 5/4, you now write P/Q ÷ R/S = P/Q · S/R. “Invert and multiply” works for polynomial fractions exactly as it does for numbers.
So the procedure is: take the reciprocal of the divisor (the second expression), change the division into multiplication, and then proceed as in any multiplication — factor, cancel, simplify. The flip is the whole new idea; everything after it you already know.
There is an extra restriction to watch for here. Because you are dividing by R/S, the expression R must not be zero either — otherwise you would be dividing by zero before flipping. So a division can introduce restrictions that come not from a written denominator but from the numerator of the divisor.
((x + 3)/x) ÷ ((x + 3)/(x - 1)) = ((x + 3)/x) · ((x - 1)/(x + 3)) = (x - 1)/x; restrictions x ≠ 0, x ≠ 1, x ≠ -3.
Flip the divisor, then multiply as usual.