1. **State the problem:** Simplify the expression $ \frac{4x^3}{2x^3 + x - 3} \cdot \frac{4x^2 + 2x - 6}{12x^3} $. 2. **Write the expression:** $$\frac{4x^3}{2x^3 + x - 3} \times \frac{4x^2 + 2x - 6}{12x^3}$$ 3. **Factor polynomials where possible:** - Factor the denominator $2x^3 + x - 3$ if possible. - Factor numerator $4x^2 + 2x - 6$. 4. **Factor $4x^2 + 2x - 6$:** $$4x^2 + 2x - 6 = 2(2x^2 + x - 3)$$ Factor quadratic inside: $$2x^2 + x - 3 = (2x - 3)(x + 1)$$ So, $$4x^2 + 2x - 6 = 2(2x - 3)(x + 1)$$ 5. **Factor denominator $2x^3 + x - 3$:** Try rational root theorem: test $x=1$: $$2(1)^3 + 1 - 3 = 2 + 1 - 3 = 0$$ So $x=1$ is a root. Divide by $x - 1$: $$\frac{2x^3 + x - 3}{x - 1} = 2x^2 + 2x + 3$$ 6. **Rewrite denominator:** $$2x^3 + x - 3 = (x - 1)(2x^2 + 2x + 3)$$ 7. **Rewrite entire expression:** $$\frac{4x^3}{(x - 1)(2x^2 + 2x + 3)} \times \frac{2(2x - 3)(x + 1)}{12x^3}$$ 8. **Multiply numerators and denominators:** $$\frac{4x^3 \times 2(2x - 3)(x + 1)}{(x - 1)(2x^2 + 2x + 3) \times 12x^3} = \frac{8x^3(2x - 3)(x + 1)}{12x^3 (x - 1)(2x^2 + 2x + 3)}$$ 9. **Cancel common factors:** Cancel $x^3$ from numerator and denominator: $$\frac{8\cancel{x^3}(2x - 3)(x + 1)}{12\cancel{x^3} (x - 1)(2x^2 + 2x + 3)}$$ Cancel common factor 4: $$\frac{\cancel{8}^2 (2x - 3)(x + 1)}{\cancel{12}^3 (x - 1)(2x^2 + 2x + 3)}$$ 10. **Final simplified expression:** $$\frac{2(2x - 3)(x + 1)}{3(x - 1)(2x^2 + 2x + 3)}$$ **Answer:** $$\boxed{\frac{2(2x - 3)(x + 1)}{3(x - 1)(2x^2 + 2x + 3)}}$$