1. **State the problem:** Simplify the expression $b \cdot ab^3 \times (-2a^5b)$ where $b = \frac{21m^2n}{7m}$. 2. **Substitute the value of $b$:** Replace $b$ in the expression with $\frac{21m^2n}{7m}$. The expression becomes $$\left(\frac{21m^2n}{7m}\right) \cdot a \left(\frac{21m^2n}{7m}\right)^3 \times \left(-2a^5 \cdot \frac{21m^2n}{7m}\right)$$ 3. **Simplify $b$: $$b = \frac{21m^2n}{7m} = \frac{\cancel{7} \times 3 m^{\cancel{1}} m n}{\cancel{7} m} = 3mn$$ 4. **Rewrite the expression using simplified $b$: $$3mn \cdot a (3mn)^3 \times (-2a^5 \cdot 3mn)$$ 5. **Calculate $(3mn)^3$: $$ (3mn)^3 = 3^3 m^3 n^3 = 27 m^3 n^3$$ 6. **Substitute back: $$3mn \cdot a \cdot 27 m^3 n^3 \times (-2 a^5 \cdot 3 mn)$$ 7. **Multiply constants and variables stepwise: $$3 \times 27 = 81$$ $$81 \times (-2) \times 3 = 81 \times (-6) = -486$$ 8. **Multiply variables: $$m^{1} \times m^{3} \times m^{1} = m^{1+3+1} = m^5$$ $$n^{1} \times n^{3} \times n^{1} = n^{1+3+1} = n^5$$ $$a^{1} \times a^{5} = a^{1+5} = a^6$$ 9. **Combine all: $$-486 a^6 m^5 n^5$$ **Final answer:** $$\boxed{-486 a^6 m^5 n^5}$$