Here's how i found the possible x-values, maybe you or someone else can figure out which is the right one without calculus.
p and q are just the x and y values (respectively) for which the line and the curve intersect.
now 24x + 3y + 2 = 0
=> 3y = -24x - 2
=> y = -8x - 2/3
now equate the functions:
since y = -8x - 2/3 and y = 1/3x³-9x
then they intersect when
-8x - 2/3 = (1/3)x³ - 9x
=>(1/3)x^3 - x + 2/3 = 0
=> x^3 - 3x + 2 = 0
we see that x = 1 is a root, so x - 1 is a factor. divide by x - 1. we obtain:
x^3 - 3x + 2 = (x - 1)(x^2 + x - 2) = (x - 1)(x + 2)(x - 1) = 0
so x = 1 or x = -2
these are the possible values for p, at one of these points we have the tangent, i just can't find a simple way to figure out which i don't want to use calculus since you're doing AS math and you're not comfortable with it
i can tell you the answer is x = p = 1, and you could just plug that value into either of the other functions to get q, but i used calculus to come to that conclusion