1. The problem is to solve the equation $\left(x - \frac{3}{2}\right) \left(\frac{3}{2}x + 2\right) = 1$. 2. Use the distributive property (FOIL) to expand the left side: $$\left(x - \frac{3}{2}\right) \left(\frac{3}{2}x + 2\right) = x \cdot \frac{3}{2}x + x \cdot 2 - \frac{3}{2} \cdot \frac{3}{2}x - \frac{3}{2} \cdot 2$$ 3. Calculate each term: $$x \cdot \frac{3}{2}x = \frac{3}{2}x^2$$ $$x \cdot 2 = 2x$$ $$- \frac{3}{2} \cdot \frac{3}{2}x = - \frac{9}{4}x$$ $$- \frac{3}{2} \cdot 2 = -3$$ 4. Combine like terms: $$\frac{3}{2}x^2 + 2x - \frac{9}{4}x - 3 = 1$$ $$\frac{3}{2}x^2 + \left(2 - \frac{9}{4}\right)x - 3 = 1$$ 5. Simplify the coefficient of $x$: $$2 - \frac{9}{4} = \frac{8}{4} - \frac{9}{4} = -\frac{1}{4}$$ 6. Rewrite the equation: $$\frac{3}{2}x^2 - \frac{1}{4}x - 3 = 1$$ 7. Subtract 1 from both sides to set the equation to zero: $$\frac{3}{2}x^2 - \frac{1}{4}x - 4 = 0$$ 8. Multiply the entire equation by 4 to clear denominators: $$4 \times \left(\frac{3}{2}x^2 - \frac{1}{4}x - 4\right) = 0$$ $$6x^2 - x - 16 = 0$$ 9. Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=6$, $b=-1$, $c=-16$: $$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 6 \times (-16)}}{2 \times 6} = \frac{1 \pm \sqrt{1 + 384}}{12} = \frac{1 \pm \sqrt{385}}{12}$$ 10. Simplify the square root if possible. Since 385 = 7 \times 55, it does not simplify nicely, so: $$x = \frac{1 \pm \sqrt{385}}{12}$$ **Final answer:** $$x = \frac{1 + \sqrt{385}}{12} \quad \text{or} \quad x = \frac{1 - \sqrt{385}}{12}$$