If `alpha, beta` be the roots of the equation `ax^2 + bx + c= 0` and `gamma, delta` those of equation `lx^2 + mx + n = 0`,then find the equation whose roots are `alphagamma+betadelta` and `alphadelta+betagamma`

Explanation

To find the equation whose roots are \( \alpha \gamma + \beta \delta \) and \( \alpha \delta + \beta \gamma \), we will follow these steps:

### Step 1: Identify the roots and their sums/products

Given the equations:
1. \( ax^2 + bx + c = 0 \) with roots \( \alpha \) and \( \beta \)
2. \( lx^2 + mx + n = 0 \) with roots \( \gamma \) and \( \delta \)

From Vieta's formulas, we know:
- For the first equation:
- \( \alpha + \beta = -\frac{b}{a} \)
- \( \alpha \beta = \frac{c}{a} \)

- For the second equation:
- \( \gamma + \delta = -\frac{m}{l} \)
- \( \gamma \delta = \frac{n}{l} \)

### Step 2: Calculate the new roots

We need to find the new roots:
1. \( r_1 = \alpha \gamma + \beta \delta \)
2. \( r_2 = \alpha \delta + \beta \gamma \)

### Step 3: Find the sum of the new roots

The sum of the new roots \( S \) is:
\[ S = r_1 + r_2 = (\alpha \gamma + \beta \delta) + (\alpha \delta + \beta \gamma) = \alpha(\gamma + \delta) + \beta(\gamma + \delta) = (\alpha + \beta)(\gamma + \delta) \]
Substituting the values from Vieta's formulas:
\[ S = \left(-\frac{b}{a}\right)\left(-\frac{m}{l}\right) = \frac{bm}{al} \]

### Step 4: Find the product of the new roots

The product of the new roots \( P \) is:
\[ P = r_1 r_2 = (\alpha \gamma + \beta \delta)(\alpha \delta + \beta \gamma) \]
Using the identity \( (x + y)(z + w) = xz + xw + yz + yw \):
\[ P = \alpha \gamma \alpha \delta + \alpha \gamma \beta \gamma + \beta \delta \alpha \delta + \beta \delta \beta \gamma \]
This can be rearranged as:
\[ P = \alpha^2 \gamma \delta + \beta^2 \gamma \delta + \alpha \beta (\gamma^2 + \delta^2) \]
Now, using \( \gamma^2 + \delta^2 = (\gamma + \delta)^2 - 2\gamma \delta \):
\[ P = \alpha^2 \gamma \delta + \beta^2 \gamma \delta + \alpha \beta \left(\left(-\frac{m}{l}\right)^2 - 2\frac{n}{l}\right) \]
Substituting the values:
\[ P = \frac{n}{l} \left(\frac{b^2}{a^2} - 2\frac{c}{a}\right) \]

### Step 5: Form the quadratic equation

The quadratic equation with roots \( r_1 \) and \( r_2 \) can be expressed as:
\[ x^2 - Sx + P = 0 \]
Substituting \( S \) and \( P \):
\[ x^2 - \frac{bm}{al} x + \frac{n}{l} \left(\frac{b^2}{a^2} - 2\frac{c}{a}\right) = 0 \]

### Final Equation

Thus, the required equation is:
\[ x^2 - \frac{bm}{al} x + \frac{n}{l} \left(\frac{b^2}{a^2} - 2\frac{c}{a}\right) = 0 \]