1. **State the problem:** Simplify the expression $$\frac{4}{-9} \times \frac{-21}{-32} \times \frac{-3}{14}$$. 2. **Recall the rule for multiplying fractions:** Multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$. 3. **Multiply the numerators:** $$4 \times (-21) \times (-3) = 4 \times (-21) = -84, \quad -84 \times (-3) = 252$$. 4. **Multiply the denominators:** $$-9 \times (-32) \times 14 = (-9) \times (-32) = 288, \quad 288 \times 14 = 4032$$. 5. **Combine the result:** $$\frac{252}{4032}$$. 6. **Simplify the fraction:** Find the greatest common divisor (GCD) of 252 and 4032. - Prime factors of 252: $$2^2 \times 3^2 \times 7$$ - Prime factors of 4032: $$2^6 \times 3^2 \times 7$$ GCD is $$2^2 \times 3^2 \times 7 = 252$$. 7. **Divide numerator and denominator by GCD:** $$\frac{\cancel{252}^{1}}{\cancel{4032}^{16}} = \frac{1}{16}$$. 8. **Final answer:** $$\boxed{\frac{1}{16}}$$.