4px2 + 12x + p is always positive

When a curve is always positive, it means that the entire curve lies above the x-axis no matter what.

This means that the curve will never cut the x-axis at all, so the above curve cannot be equated to zero at all (ie no real roots).

In addition, the curve must be a smiley face curve, where the coefficient of x2 must be positive.

Condition 1 is that the coefficient of x2 > 0, so 4p > 0, or p > 0.

Condition 2 is that the discriminant of roots < 0.

b2 - 4ac < 0
144 - 4 (4p) (p) < 0
144 - 4p2 < 0
36 - p2 < 0
p2 - 36 > 0
(p + 6) (p - 6) > 0

Solving for p,

p < -6 or p > 6

We reflect the case p < -6 as this interferes with our first condition that p must be positive.

So, p > 6.

The least possible integer of p must be 7.

Thanks for explaining so clearly.