1. **State the problem:** Simplify the expression $$\frac{4x}{2x^2 + x - 3} \cdot \frac{4x^2 + 2x - 6}{12x^3}$$. 2. **Factor all polynomials where possible:** - Factor the quadratic in the denominator of the first fraction: $$2x^2 + x - 3 = (2x - 3)(x + 1)$$ - Factor the numerator of the second fraction: $$4x^2 + 2x - 6 = 2(2x^2 + x - 3) = 2(2x - 3)(x + 1)$$ 3. **Rewrite the expression with factored forms:** $$\frac{4x}{(2x - 3)(x + 1)} \cdot \frac{2(2x - 3)(x + 1)}{12x^3}$$ 4. **Multiply the numerators and denominators:** $$\frac{4x \cdot 2(2x - 3)(x + 1)}{(2x - 3)(x + 1) \cdot 12x^3}$$ 5. **Cancel common factors:** - Cancel $(2x - 3)$ and $(x + 1)$ from numerator and denominator: $$\frac{4x \cdot 2 \cancel{(2x - 3)} \cancel{(x + 1)}}{\cancel{(2x - 3)} \cancel{(x + 1)} \cdot 12x^3} = \frac{8x}{12x^3}$$ 6. **Simplify the fraction:** $$\frac{8x}{12x^3} = \frac{\cancel{8}^2 \cancel{x}}{\cancel{12}^3 x^3} = \frac{2}{3x^2}$$ 7. **Final answer:** $$\boxed{\frac{2}{3x^2}}$$