1. **State the problem:** Simplify the expression $$\frac{2}{3}ax \left(-2ax^2\right) - \left(-\frac{1}{6}a^2\right) x^3 + \left(-\frac{1}{2}a^2 x\right) 3x + \frac{3}{4}a^2 x (2x)$$. 2. **Apply multiplication and distribute terms:** - Multiply $$\frac{2}{3}ax$$ and $$-2ax^2$$: $$\frac{2}{3}ax \times -2ax^2 = \frac{2}{3} \times -2 \times a \times a \times x \times x^2 = -\frac{4}{3}a^2 x^3$$ - The second term is $$-\left(-\frac{1}{6}a^2\right) x^3 = +\frac{1}{6}a^2 x^3$$ - Multiply $$-\frac{1}{2}a^2 x$$ and $$3x$$: $$-\frac{1}{2}a^2 x \times 3x = -\frac{3}{2}a^2 x^2$$ - Multiply $$\frac{3}{4}a^2 x$$ and $$2x$$: $$\frac{3}{4}a^2 x \times 2x = \frac{3}{4} \times 2 \times a^2 \times x \times x = \frac{3}{2}a^2 x^2$$ 3. **Rewrite the expression with simplified terms:** $$-\frac{4}{3}a^2 x^3 + \frac{1}{6}a^2 x^3 - \frac{3}{2}a^2 x^2 + \frac{3}{2}a^2 x^2$$ 4. **Combine like terms:** - Combine $$x^3$$ terms: $$-\frac{4}{3}a^2 x^3 + \frac{1}{6}a^2 x^3 = \left(-\frac{4}{3} + \frac{1}{6}\right) a^2 x^3$$ Find common denominator 6: $$-\frac{8}{6} + \frac{1}{6} = -\frac{7}{6}$$ So, $$-\frac{7}{6}a^2 x^3$$ - Combine $$x^2$$ terms: $$-\frac{3}{2}a^2 x^2 + \frac{3}{2}a^2 x^2 = 0$$ 5. **Final simplified expression:** $$-\frac{7}{6}a^2 x^3$$ **Answer:** $$-\frac{7}{6}a^2 x^3$$