In our case, we can note that the slope m is the coefficient of the variable x, that is,
[tex]m=4[/tex]
A perpendicular line must have a negative reciprocal slope, that is,
[tex]\begin{gathered} M=-\frac{1}{m} \\ then \\ M=-\frac{1}{4} \end{gathered}[/tex]
then, the searched line has the form
[tex]y=-\frac{1}{4}x+b[/tex]
Now, we can find b by means of the given point (4,-2). Then, by replacing this point into the last equation, we get
[tex]-2=-\frac{1}{4}(4)+b[/tex]
which gives
[tex]-2=-1+b[/tex]
and by moving -1 to the left hand side, we have
[tex]\begin{gathered} -2+1=b \\ -1=b \\ or\text{ quivalently,} \\ b=-1 \end{gathered}[/tex]
Therefore, the perpendicular line is
[tex]y=-\frac{1}{4}x-1[/tex]
which corresponds to option b
