求 $p$ 的取值范围，使得曲线 $y = px^2 + 2(p + 2)x + p + 7$ 与 $x$ 轴没有交点。

1. State the condition for the curve to have no $x$-intercepts

For the curve to have no $x$-intercepts, the equation $px^2 + 2(p + 2)x + p + 7 = 0$ must have no real roots. Therefore, the discriminant must be less than zero:
$b^2 - 4ac < 0$

2. Substitute the coefficients into the discriminant formula

$[2(p + 2)]^2 - 4(p)(p + 7) < 0$

3. Expand and simplify the inequality

$4(p^2 + 4p + 4) - 4p^2 - 28p < 0$
$4p^2 + 16p + 16 - 4p^2 - 28p < 0$
$-12p + 16 < 0$

4. Solve the linear inequality for $p$

$16 < 12p$
$p > \frac{16}{12}$
$p > \frac{4}{3}$