If 'a' is an integer lying in $[-5, 30]$, then the probability that the graph of $y = x^2 + 2(a + 4)x - 5a + 64$ lies above the x-axis is

For the graph of the quadratic polynomial $y = x^2 + 2(a + 4)x - 5a + 64$ to lie strictly above the x-axis, the coefficient of $x^2$ must be positive and its discriminant $D$ must be strictly less than zero.

The coefficient of $x^2$ is $1$, which is positive.

Setting the discriminant $D < 0$:

$[2(a + 4)]^2 - 4(1)(-5a + 64) < 0$

$4(a^2 + 8a + 16) + 20a - 256 < 0$

$4a^2 + 32a + 64 + 20a - 256 < 0$

$4a^2 + 52a - 192 < 0$

Dividing the entire inequality by $4$:

$a^2 + 13a - 48 < 0$

Factoring the quadratic expression:

$(a + 16)(a - 3) < 0$

This gives the range of $a$ as:

$-16 < a < 3$

It is given that $a$ is an integer in the interval $[-5, 30]$.

The total number of possible integer values for $a$ in $[-5, 30]$ is $30 - (-5) + 1 = 36$.

The favorable integer values of $a$ must satisfy both $-16 < a < 3$ and $a \in [-5, 30]$. The integers satisfying both conditions are $-5, -4, -3, -2, -1, 0, 1, 2$.

The number of favorable outcomes is $8$.

The required probability is the ratio of the number of favorable outcomes to the total number of outcomes:

$\text{Probability} = \dfrac{8}{36} = \dfrac{2}{9}$

Answer: $\dfrac{2}{9}$