1. **State the problem:** Simplify the expression $$2x\left(2(\ln(x)^2)-3\ln(x)+2\right)-\frac{7x}{4}$$. 2. **Recall the distributive property:** To simplify, distribute $2x$ across the terms inside the parentheses. 3. **Apply distribution:** $$2x \times 2(\ln(x)^2) = 4x(\ln(x)^2)$$ $$2x \times (-3\ln(x)) = -6x\ln(x)$$ $$2x \times 2 = 4x$$ 4. **Rewrite the expression:** $$4x(\ln(x)^2) - 6x\ln(x) + 4x - \frac{7x}{4}$$ 5. **Combine like terms:** The terms $4x$ and $-\frac{7x}{4}$ are like terms. Convert $4x$ to quarters: $$4x = \frac{16x}{4}$$ So, $$\frac{16x}{4} - \frac{7x}{4} = \frac{9x}{4}$$ 6. **Final simplified expression:** $$4x(\ln(x)^2) - 6x\ln(x) + \frac{9x}{4}$$ This is the simplified form of the original expression. **Answer:** $$4x(\ln(x)^2) - 6x\ln(x) + \frac{9x}{4}$$