1. **State the problem:** Simplify the expression $$\frac{2x^3 + 4x^2}{3x^2 + 2x}$$. 2. **Write the formula and rules:** To simplify a rational expression, factor numerator and denominator and then cancel common factors. 3. **Factor numerator:** $$2x^3 + 4x^2 = 2x^2(x + 2)$$. 4. **Factor denominator:** $$3x^2 + 2x = x(3x + 2)$$. 5. **Rewrite the expression:** $$\frac{2x^2(x + 2)}{x(3x + 2)}$$. 6. **Cancel common factor $x$:** $$\frac{2\cancel{x}x(x + 2)}{\cancel{x}(3x + 2)} = \frac{2x(x + 2)}{3x + 2}$$. 7. **Final simplified form:** $$\frac{2x(x + 2)}{3x + 2}$$. This is the simplest form unless further factorization is possible, which it is not here.