1. The problem asks for the slope of a line perpendicular to the line given by the equation $$y = 4x + \frac{3}{2}$$. 2. Recall that the slope-intercept form of a line is $$y = mx + b$$, where $$m$$ is the slope. 3. From the given equation, the slope $$m$$ of the original line is $$4$$. 4. Important rule: The slope of a line perpendicular to another line is the negative reciprocal of the original slope. 5. The negative reciprocal of $$4$$ is $$-\frac{1}{4}$$. 6. Therefore, the slope of the line perpendicular to $$y = 4x + \frac{3}{2}$$ is $$-\frac{1}{4}$$. Final answer: $$-\frac{1}{4}$$.