Answer
$$ y=-\frac{1}{4}x+\frac{19}{4}.$$
Work Step by Step
Since the line is perpendicular to the line given by $ y = 4x + 2$ then its slope is $-\frac{1}{4}$, and hence the line has the equation
$$ y=-\frac{1}{4}x+c.$$
To find $ c $, we make use of the fact that the line passes through $(3,4)$:
$$4=-\frac{1}{4}(3)+c\Longrightarrow c=\frac{19}{4}$$
Hence the equation of the line is
$$ y=-\frac{1}{4}x+\frac{19}{4}.$$
