1. **State the problem:** Find the equation of the line passing through the points $\left(-\frac{3}{2}, 4\right)$ and $\left(\frac{5}{2}, -2\right)$.\n\n2. **Formula used:** The slope $m$ of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}.$$\nThe equation of the line in point-slope form is $$y - y_1 = m(x - x_1).$$\n\n3. **Calculate the slope:**\n$$m = \frac{-2 - 4}{\frac{5}{2} - \left(-\frac{3}{2}\right)} = \frac{-6}{\frac{5}{2} + \frac{3}{2}} = \frac{-6}{\frac{8}{2}} = \frac{-6}{4} = -\frac{3}{2}.$$\n\n4. **Write the equation using point-slope form with point $\left(-\frac{3}{2}, 4\right)$:**\n$$y - 4 = -\frac{3}{2}\left(x - \left(-\frac{3}{2}\right)\right) = -\frac{3}{2}\left(x + \frac{3}{2}\right).$$\n\n5. **Simplify the equation:**\n$$y - 4 = -\frac{3}{2}x - \frac{3}{2} \times \frac{3}{2} = -\frac{3}{2}x - \frac{9}{4}.$$\n\n6. **Add 4 to both sides:**\n$$y = -\frac{3}{2}x - \frac{9}{4} + 4.$$\n\n7. **Convert 4 to quarters:**\n$$4 = \frac{16}{4},$$\nso\n$$y = -\frac{3}{2}x - \frac{9}{4} + \frac{16}{4} = -\frac{3}{2}x + \frac{7}{4}.$$\n\n**Final answer:** $$y = -\frac{3}{2}x + \frac{7}{4}.$$