We have to find the slope of the given line. We can solve its equation for y to get
[tex]2x-3y+6=0\iff 3y=2x+6 \iff y = \dfrac{2}{3}y+2[/tex]
So, the slope of the given line is 2/3.
If m and m' are the slopes of two perpendicular lines, we have
[tex]mm'=-1[/tex]
This means that the perpendicular slope is -3/2.
Now we use the point-slope formula
[tex]y-y_0=m(x-x_0)[/tex]
where [tex](x_0,y_0)[/tex] is the given point and m is the given slope, to find the equation of the required line:
[tex]y-3=-\dfrac{3}{2}(x-4) \iff y = -\dfrac{3}{2}x+9[/tex]
