The x = y^2 -4 is a horizontal parabola that opens to the right. Its vertex is at (-4,0).

The line L intersects the parabola at (0,p) and (5,t).

Point (0,p) is (0,2) or (0,-2).

Point (5,t) is (5,3) or (5,-3).

There are only two possible L with positive slope.

The L with the greatest possible value of slope is the L that passes through (0,-2) and (5,3)

m = (-2 -3) / (0 -5) = 1

Check:

The slope of the L that passes through ((0,2) and (5,3) is

m = (2 -3) / (0 -5) = 1/5 --------less than the 1 of the L above.

The L that passes through (0,2) and (5,-3) has a negative value slope.