Let the set of all values of $p \in \mathbb{R}$, for which both the roots of the equation $x^2 - (p + 2)x + (2p + 9) = 0$ are negative real numbers, be the interval $(\alpha, \beta]$. Then $\beta - 2\alpha$ is equal to:

Step 1: Identify Conditions for Negative Real Roots

For a quadratic equation $ax^2 + bx + c = 0$ to have two negative real roots, three conditions must be satisfied. First, the discriminant $D$ must be greater than or equal to zero for real roots. Second, the sum of the roots must be negative. Third, the product of the roots must be positive.

Step 2: Apply Discriminant Condition

The discriminant $D$ is calculated as $b^2 - 4ac$. Substituting the coefficients from the given equation, we get $(-(p+2))^2 - 4(1)(2p+9) \ge 0$. Simplifying this inequality leads to $p^2 - 4p - 32 \ge 0$, which factors into $(p-8)(p+4) \ge 0$. This inequality holds when $p \le -4$ or $p \ge 8$.

Step 3: Apply Sum of Roots Condition

The sum of the roots is given by $-b/a$. For negative roots, this sum must be less than zero. Substituting the coefficients, we have $-(-(p+2))/1 < 0$, which simplifies to $p+2 < 0$. Therefore, $p$ must be less than $-2$.

Step 4: Apply Product of Roots Condition

The product of the roots is given by $c/a$. For negative roots, this product must be greater than zero. Substituting the coefficients, we have $(2p+9)/1 > 0$, which simplifies to $2p+9 > 0$. This implies $2p > -9$, so $p > -9/2$ or $p > -4.5$.

Step 5: Find Intersection of Conditions

To find the values of $p$ that satisfy all three conditions, we need to find the intersection of the three intervals. The first condition gives $p \in (-\infty, -4] \cup [8, \infty)$. The second condition gives $p \in (-\infty, -2)$. The third condition gives $p \in (-4.5, \infty)$. The common interval for all three conditions is $(-4.5, -4]$.

Step 6: Calculate $\beta - 2\alpha$

From the intersection of the conditions, we found that $p$ belongs to the interval $(-4.5, -4]$. Comparing this with the given interval $(\alpha, \beta]$, we have $\alpha = -4.5$ and $\beta = -4$. Now, we can calculate $\beta - 2\alpha$ by substituting these values: $-4 - 2(-4.5) = -4 + 9 = 5$.

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