1. **State the problem:** Simplify the expression $$\frac{7x^2 - 4x + 3}{2x} \times \frac{x}{x - 3}$$ and solve for $$x$$. 2. **Rewrite the expression:** $$\frac{7x^2 - 4x + 3}{2x} \times \frac{x}{x - 3} = \frac{7x^2 - 4x + 3}{2x} \cdot \frac{x}{x - 3}$$ 3. **Simplify by canceling common factors:** The $$x$$ in numerator and denominator cancels out: $$= \frac{7x^2 - 4x + 3}{2(x - 3)}$$ 4. **Factor the numerator if possible:** Try to factor $$7x^2 - 4x + 3$$. Check discriminant $$\Delta = (-4)^2 - 4 \cdot 7 \cdot 3 = 16 - 84 = -68 < 0$$, so no real factors. 5. **Final simplified expression:** $$\frac{7x^2 - 4x + 3}{2(x - 3)}$$ 6. **Solve for $$x$$ if equation is given as:** $$\frac{7x^2 - 4x + 3}{2(x - 3)} = x$$ Multiply both sides by $$2(x - 3)$$: $$7x^2 - 4x + 3 = 2x(x - 3)$$ Expand right side: $$7x^2 - 4x + 3 = 2x^2 - 6x$$ Bring all terms to one side: $$7x^2 - 4x + 3 - 2x^2 + 6x = 0$$ Simplify: $$5x^2 + 2x + 3 = 0$$ 7. **Solve quadratic equation:** Use quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=5$$, $$b=2$$, $$c=3$$. Calculate discriminant: $$\Delta = 2^2 - 4 \cdot 5 \cdot 3 = 4 - 60 = -56$$ Since $$\Delta < 0$$, solutions are complex: $$x = \frac{-2 \pm \sqrt{-56}}{10} = \frac{-2 \pm \sqrt{56}i}{10}$$ Simplify $$\sqrt{56} = \sqrt{4 \cdot 14} = 2\sqrt{14}$$: $$x = \frac{-2 \pm 2\sqrt{14}i}{10} = \frac{-1 \pm \sqrt{14}i}{5}$$ **Final answers:** $$x = \frac{-1 + \sqrt{14}i}{5}, \frac{-1 - \sqrt{14}i}{5}$$