Step 1: Understanding the Concept:
A quadratic graph \( y = ax^2 + bx + c \) lies entirely above the x-axis if \( a>0 \) and the discriminant \( D<0 \). Here, \( a = 1>0 \), so we only need to ensure \( D<0 \).
Step 2: Key Formula or Approach:
1. Total outcomes: Count of integers in \( [-5, 30] \).
2. Favorable outcomes: Find integers satisfying \( D<0 \).
Step 3: Detailed Explanation:
Total integers \( = 30 - (-5) + 1 = 36 \).
Condition for \( D<0 \):
\[ [2(a+4)]^2 - 4(1)(64 - 5a)<0 \]
\[ 4(a+4)^2 - 4(64 - 5a)<0 \]
\[ a^2 + 8a + 16 - 64 + 5a<0 \]
\[ a^2 + 13a - 48<0 \]
Factorizing the quadratic \( a^2 + 13a - 48 \):
Product = -48, Sum = 13. Factors are \( 16 \) and \( -3 \).
\[ (a + 16)(a - 3)<0 \]
This inequality holds for \( a \in (-16, 3) \).
Now, we find integers in this range that also lie in the given interval \( [-5, 30] \):
The common range is \( a \in [-5, 3) \).
Integers are: \( -5, -4, -3, -2, -1, 0, 1, 2 \).
Number of favorable integers \( = 2 - (-5) + 1 = 8 \).
Probability \( P = \frac{\text{Favorable}}{\text{Total}} = \frac{8}{36} \).
Simplifying the fraction:
\[ P = \frac{2}{9} \]
Step 4: Final Answer:
The probability is \( 2/9 \), which corresponds to option (C).