If `alpha and beta` are the roots of the equation `x^(2) - q(1+x) - r =0 , " then what is " (1+ alpha) (1+ beta)` equal to ?

Options

A. `1-r`
B. `q-r`
C. `1+r`
D. `q+r`

Explanation

To find the value of \( (1 + \alpha)(1 + \beta) \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 - q(1 + x) - r = 0 \), we can follow these steps:

### Step 1: Rewrite the Equation
The given equation is:
\[ x^2 - q(1 + x) - r = 0 \]
We can rewrite it as:
\[ x^2 - qx - q - r = 0 \]
This shows that the coefficients of the polynomial are:
- \( a = 1 \)
- \( b = -q \)
- \( c = -q - r \)

### Step 2: Use Vieta's Formulas
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-q}{1} = q \]
- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{-q - r}{1} = -q - r \]

### Step 3: Calculate \( (1 + \alpha)(1 + \beta) \)
Now, we can expand \( (1 + \alpha)(1 + \beta) \):
\[ (1 + \alpha)(1 + \beta) = 1 + \alpha + \beta + \alpha \beta \]
Substituting the values we found:
\[ = 1 + (q) + (-q - r) \]
This simplifies to:
\[ = 1 + q - q - r = 1 - r \]

### Final Result
Thus, the value of \( (1 + \alpha)(1 + \beta) \) is:
\[ \boxed{1 - r} \]