To solve the problem, we need to determine the conditions under which the quadratic expression \( x^2 + 2(a + 4)x - 5a + 64 > 0 \) holds for all \( x \in \mathbb{R} \). This is equivalent to ensuring that the quadratic has no real roots, which occurs when its discriminant is less than zero.
### Step-by-Step Solution:
1. **Identify the coefficients of the quadratic**:
The given quadratic can be rewritten in standard form \( Ax^2 + Bx + C \), where:
- \( A = 1 \)
- \( B = 2(a + 4) = 2a + 8 \)
- \( C = -5a + 64 \)
2. **Calculate the discriminant**:
The discriminant \( D \) of a quadratic \( Ax^2 + Bx + C \) is given by:
\[
D = B^2 - 4AC
\]
Plugging in our coefficients:
\[
D = (2a + 8)^2 - 4(1)(-5a + 64)
\]
3. **Expand the discriminant**:
\[
D = (2a + 8)^2 + 20a - 256
\]
Expanding \( (2a + 8)^2 \):
\[
D = 4a^2 + 32a + 64 + 20a - 256
\]
Simplifying:
\[
D = 4a^2 + 52a - 192
\]
4. **Set the discriminant less than zero**:
To ensure the quadratic is always positive, we need:
\[
4a^2 + 52a - 192 < 0
\]
5. **Solve the quadratic inequality**:
First, we find the roots of the equation \( 4a^2 + 52a - 192 = 0 \) using the quadratic formula:
\[
a = \frac{-B \pm \sqrt{D}}{2A} = \frac{-52 \pm \sqrt{52^2 - 4 \cdot 4 \cdot (-192)}}{2 \cdot 4}
\]
Calculating the discriminant:
\[
D = 2704 + 3072 = 5776
\]
Thus,
\[
a = \frac{-52 \pm 76}{8}
\]
This gives us:
\[
a_1 = \frac{24}{8} = 3 \quad \text{and} \quad a_2 = \frac{-128}{8} = -16
\]
6. **Determine the intervals**:
The roots are \( a = -16 \) and \( a = 3 \). The quadratic opens upwards (since the coefficient of \( a^2 \) is positive), so the inequality \( 4a^2 + 52a - 192 < 0 \) holds between the roots:
\[
-16 < a < 3
\]
7. **Consider the range of \( a \)**:
The problem states that \( a \) is selected from the interval \([-5, 30]\). We need to find the intersection of the intervals:
\[
[-5, 30] \cap (-16, 3) = [-5, 3)
\]
8. **Count the integers in the interval**:
The integers in the interval \([-5, 3)\) are:
\(-5, -4, -3, -2, -1, 0, 1, 2\).
This gives us a total of 8 integers.
9. **Calculate the total number of integers in the range \([-5, 30]\)**:
The total integers from \(-5\) to \(30\) inclusive are:
\[
30 - (-5) + 1 = 36
\]
10. **Calculate the probability**:
The probability \( P \) is given by the ratio of favorable outcomes to total outcomes:
\[
P = \frac{8}{36} = \frac{2}{9}
\]
### Final Answer:
Thus, the probability of selecting integers \( a \) from the interval \([-5, 30]\) such that the quadratic is positive for all \( x \in \mathbb{R} \) is \( \frac{2}{9} \).
