Both roots of equation `x^(2) - (a + 3)x + a + 4 = 0` are negative. Calculate the values of a.

Answer

To solve the problem, we need to determine the values of \( a \) such that both roots of the quadratic equation \( x^2 - (a + 3)x + (a + 4) = 0 \) are negative.

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c \). Here:
- \( a = 1 \)
- \( b = -(a + 3) \)
- \( c = a + 4 \)

### Step 2: Use the conditions for the roots
For both roots of the quadratic equation to be negative, we need to satisfy two conditions:
1. The sum of the roots \( \alpha + \beta \) must be negative.
2. The product of the roots \( \alpha \beta \) must be positive.

### Step 3: Apply the sum of roots condition
The sum of the roots can be expressed as:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-(a + 3)}{1} = a + 3 \]
For the sum to be negative:
\[ a + 3 < 0 \implies a < -3 \]

### Step 4: Apply the product of roots condition
The product of the roots can be expressed as:
\[ \alpha \beta = \frac{c}{a} = \frac{a + 4}{1} = a + 4 \]
For the product to be positive:
\[ a + 4 > 0 \implies a > -4 \]

### Step 5: Combine the inequalities
Now we have two inequalities:
1. \( a < -3 \)
2. \( a > -4 \)

Combining these inequalities gives us:
\[ -4 < a < -3 \]

### Conclusion
Thus, the values of \( a \) for which both roots of the equation are negative are:
\[ a \in (-4, -3) \]