33. \( \frac{3 m^{2}-5 m n+2 n^{2}}{9 m^{2}-4 n^{2}} \div \frac{m^{3}-m^{2} n}{9 m^{2}+6 m n} \)

We begin with the expression   [(3m² – 5mn + 2n²) / (9m² – 4n²)] ÷ [(m³ – m²n) / (9m² + 6mn)]. Step 1. Factor each part. • Factor the numerator of the first fraction:   3m² – 5mn + 2n²   = (3m – 2n)(m – n). • Factor the denominator of the first fraction, recognizing a difference of squares:   9m² – 4n² = (3m – 2n)(3m + 2n). Thus, the first fraction becomes   [(3m – 2n)(m – n)] / [(3m – 2n)(3m + 2n)]. Assuming 3m – 2n ≠ 0, cancel the common factor to get:   (m – n) / (3m + 2n). Step 2. Simplify the second fraction. • Factor the numerator:   m³ – m²n = m²(m – n). • Factor the denominator by factoring out 3m:   9m² + 6mn = 3m(3m + 2n). Thus, the second fraction becomes   [m²(m – n)] / [3m(3m + 2n)]. Cancel one factor of m (assuming m ≠ 0) to get:   m(m – n) / [3(3m + 2n)]. Step 3. Set up the division. Divide the simplified first fraction by the simplified second fraction:   [(m – n) / (3m + 2n)] ÷ [m(m – n) / {3(3m + 2n)}]. Dividing by a fraction is equivalent to multiplying by its reciprocal:   = [(m – n) / (3m + 2n)] × [3(3m + 2n) / (m(m – n))]. Step 4. Cancel common factors. The factors (m – n) and (3m + 2n) cancel (provided m – n ≠ 0 and 3m + 2n ≠ 0):   = 3 / m. Thus, the simplified expression is   3/m. Remember, the cancellation steps require that 3m – 2n, m – n, 3m + 2n, and m are nonzero. Final Answer: 3/m.