1. **State the problem:** Simplify the expression $$\frac{-12xy}{7y^4} \cdot \frac{21x^5y^2}{4y}$$. 2. **Write the multiplication of fractions:** $$\frac{-12xy}{7y^4} \times \frac{21x^5y^2}{4y} = \frac{-12xy \cdot 21x^5y^2}{7y^4 \cdot 4y}$$ 3. **Multiply numerators and denominators:** $$\frac{-12 \times 21 \times x \times x^5 \times y \times y^2}{7 \times 4 \times y^4 \times y}$$ 4. **Simplify coefficients:** $$-12 \times 21 = -252$$ $$7 \times 4 = 28$$ So, $$\frac{-252 x^{1+5} y^{1+2}}{28 y^{4+1}} = \frac{-252 x^6 y^3}{28 y^5}$$ 5. **Simplify the fraction of coefficients:** $$\frac{-252}{28} = \frac{\cancel{-252}^{9}}{\cancel{28}^{1}} = -9$$ 6. **Simplify the powers of $y$ using the rule $\frac{y^a}{y^b} = y^{a-b}$:** $$\frac{y^3}{y^5} = y^{3-5} = y^{-2} = \frac{1}{y^2}$$ 7. **Combine all simplified parts:** $$-9 x^6 \times \frac{1}{y^2} = \frac{-9 x^6}{y^2}$$ **Final answer:** $$\boxed{\frac{-9 x^6}{y^2}}$$