1. **State the problem:** Simplify the expression $$(b x^4 - 3 b^2 x^3 + 4 b^4 x - b)(-2 b^3 x^2).$$ 2. **Recall the distributive property:** To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial. 3. **Multiply each term:** - Multiply $b x^4$ by $-2 b^3 x^2$: $$b \cdot (-2 b^3) x^{4+2} = -2 b^{1+3} x^6 = -2 b^4 x^6.$$ - Multiply $-3 b^2 x^3$ by $-2 b^3 x^2$: $$-3 b^2 \cdot (-2 b^3) x^{3+2} = 6 b^{2+3} x^5 = 6 b^5 x^5.$$ - Multiply $4 b^4 x$ by $-2 b^3 x^2$: $$4 b^4 \cdot (-2 b^3) x^{1+2} = -8 b^{4+3} x^3 = -8 b^7 x^3.$$ - Multiply $-b$ by $-2 b^3 x^2$: $$-b \cdot (-2 b^3) x^2 = 2 b^{1+3} x^2 = 2 b^4 x^2.$$ 4. **Write the final simplified expression:** $$-2 b^4 x^6 + 6 b^5 x^5 - 8 b^7 x^3 + 2 b^4 x^2.$$