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

Options

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

Explanation

To solve the problem, we need to find the value of \((\beta + \lambda)(\beta + \delta)\) given the roots of two quadratic equations.

### Step-by-Step Solution:

1. **Identify the first quadratic equation**:
The first equation is given as:
\[ x^2 + px - q = 0 \]
The roots of this equation are \(\alpha\) and \(\beta\).

2. **Use Vieta's formulas for the first equation**:
By Vieta's formulas, we know:
- The sum of the roots \(\alpha + \beta = -\frac{b}{a} = -\frac{p}{1} = -p\)
- The product of the roots \(\alpha \beta = \frac{-q}{1} = -q\)

3. **Identify the second quadratic equation**:
The second equation is given as:
\[ x^2 - px + r = 0 \]
The roots of this equation are \(\lambda\) and \(\delta\).

4. **Use Vieta's formulas for the second equation**:
Again, applying Vieta's formulas:
- The sum of the roots \(\lambda + \delta = -\frac{-p}{1} = p\)
- The product of the roots \(\lambda \delta = \frac{r}{1} = r\)

5. **Expand \((\beta + \lambda)(\beta + \delta)\)**:
We want to find:
\[ (\beta + \lambda)(\beta + \delta) = \beta^2 + \beta\delta + \beta\lambda + \lambda\delta \]

6. **Substitute known values**:
From Vieta's formulas:
- We know \(\lambda + \delta = p\)
- We know \(\lambda \delta = r\)

Thus, we can rewrite the expression:
\[ = \beta^2 + \beta(\lambda + \delta) + \lambda\delta \]
\[ = \beta^2 + \beta p + r \]

7. **Find \(\beta^2\)**:
We can express \(\beta^2\) using the first equation. From the first equation:
\[ \beta^2 = -p\beta - q \]

8. **Substitute \(\beta^2\) into the expression**:
Substituting \(\beta^2\) into our earlier expression:
\[ = (-p\beta - q) + \beta p + r \]
The \(\beta p\) terms cancel out:
\[ = -q + r \]

9. **Final expression**:
Therefore, we have:
\[ (\beta + \lambda)(\beta + \delta) = r - q \]

### Conclusion:
The value of \((\beta + \lambda)(\beta + \delta)\) is:
\[ \boxed{r - q} \]