The equation of line that passes through $(-1,3)$ and is perpendicular to the straight line $5 x+3 y=-1$ is:

$3 x-5 y+18=0$

The given line

5 x + 3 y + 1 = 0

can be rewritten as

3 y = -5 x + 1

or

y = (-5/3) x - 1/3 —————————————-(1)

has a slope = -5/3

The slope of the straight line perpendicular to the given line =  3/5

The equation of straight line with slope m and passing through the point (x¹, y¹) is

y - y¹ = m (x - x¹).

In our case (x¹, y¹) = (-1, 3) and the equation of the required line becomes

y - (3) = (3/5) (x –(-1))

=> y - 3 =  3 x/5 + 3/5

=> 5 y - 15 =  3 x + 3

=> 3 x - 5 y + 18  = 0

The required equation is:

3 x - 5 y + 18  = 0.