If the point (1,4) lies inside the circle x² + y² - 6x - 10y + p = 0 and the circle does not touch or intersect the coordinate axes, then the set of all possible values of p is the interval:

The equation of the circle is given by x² + y² - 6x - 10y + p = 0.
We can rewrite this equation in the standard form (x - a)² + (y - b)² = r², where (a, b) is the center and r is the radius of the circle.
Completing the square, we get:
(x² - 6x) + (y² - 10y) + p = 0
(x² - 6x + 9) + (y² - 10y + 25) + p - 9 - 25 = 0
(x - 3)² + (y - 5)² = 34 - p
Since the circle does not touch or intersect the coordinate axes, the distance from the center (3, 5) to the origin (0, 0) must be greater than the radius r. The distance from the center to the origin is √(3² + 5²) = √34.
Therefore, √34 > √(34 - p), which implies 34 > 34 - p, so p > 0.
Also, the point (1, 4) lies inside the circle, so the distance from (1, 4) to the center (3, 5) must be less than the radius r.
The distance from (1, 4) to (3, 5) is √((3 - 1)² + (5 - 4)²) = √(2² + 1²) = √5.
Therefore, √5 < √(34 - p), which implies 5 < 34 - p, so p < 29.
Combining these conditions, we have 0 < p < 29. Thus, the set of all possible values of p is the interval (0, 29).