Write the result after applying componendo and dividendo to `3a^(2)+2b^(2):3a^(2)-2b^(2): :3c^(2)+2d^(2):3c^(2)-2d^(2)`.

Answer

To solve the problem using the method of componendo and dividendo, we will follow these steps:

### Step 1: Set Up the Ratios
We start with the given proportion:
\[ \frac{3a^2 + 2b^2}{3a^2 - 2b^2} = \frac{3c^2 + 2d^2}{3c^2 - 2d^2} \]

### Step 2: Apply Componendo and Dividendo
According to the method of componendo and dividendo, we add the numerator and denominator of each ratio and also subtract the denominator from the numerator.

For the left side:
- **Numerator**: \( (3a^2 + 2b^2) + (3a^2 - 2b^2) = 6a^2 \)
- **Denominator**: \( (3a^2 - 2b^2) - (3a^2 + 2b^2) = -4b^2 \)

So, the left side becomes:
\[ \frac{6a^2}{-4b^2} \]

For the right side:
- **Numerator**: \( (3c^2 + 2d^2) + (3c^2 - 2d^2) = 6c^2 \)
- **Denominator**: \( (3c^2 - 2d^2) - (3c^2 + 2d^2) = -4d^2 \)

So, the right side becomes:
\[ \frac{6c^2}{-4d^2} \]

### Step 3: Write the New Proportion
Now we can write the new proportion:
\[ \frac{6a^2}{-4b^2} = \frac{6c^2}{-4d^2} \]

### Step 4: Simplify the Proportion
We can simplify this proportion by canceling out the common factors:
\[ \frac{6}{-4} \cdot \frac{a^2}{b^2} = \frac{6}{-4} \cdot \frac{c^2}{d^2} \]

This leads to:
\[ \frac{a^2}{b^2} = \frac{c^2}{d^2} \]

### Step 5: Cross Multiply
Cross multiplying gives us:
\[ a^2 d^2 = b^2 c^2 \]

### Step 6: Rearranging the Equation
Rearranging the equation gives:
\[ a^2 d^2 - b^2 c^2 = 0 \]

### Final Result
Thus, the final result after applying componendo and dividendo is:
\[ a^2 d^2 + b^2 c^2 = 0 \]