Triangle

According to the triangle inequality....we have the following system of equations

(3p - 1) + 3p >  (p^2 + 1 )  →   6p - 1 > p^2 + 1    →  p^2 - 6p  + 2   <  0    (1)

(p^2 + 1) + 3p > 3p - 1    →    p^2   > - 2      (2)

p^2 + 1 + 3p - 1 >   3p    →   p^2    >  0   (3)

The last two are true for any positive integer p, so we can ignore  them

Looking at the first equation and setting it to 0 , we have that

p^2 - 6p + 2  = 0        the solutions (roots) to this equation   are    p ≈ .35  and  p ≈ 5.6

And since this is an upward turning parabola, all  integers between these two values will make (1) true...

So.....the  integers that make  this true are  1,2,3,4 5

So.....there are five integer values for  p