The perimeter of a triangle is `6p^(2) - 4 p + 9` and two of its sides are `p^(2) - 2p + 1` and `3p^(2) - 5p + 3`. Find the third side of the triangle.

Answer

To find the third side of the triangle given the perimeter and the lengths of two sides, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given information**:
- Perimeter of the triangle: \( P = 6p^2 - 4p + 9 \)
- First side: \( A = p^2 - 2p + 1 \)
- Second side: \( B = 3p^2 - 5p + 3 \)

2. **Set up the equation for the perimeter**:
The perimeter of a triangle is the sum of its three sides:
\[ P = A + B + C \]
where \( C \) is the third side we need to find.

3. **Substitute the known values into the equation**:
\[ 6p^2 - 4p + 9 = (p^2 - 2p + 1) + (3p^2 - 5p + 3) + C \]

4. **Combine the expressions for sides A and B**:
\[ A + B = (p^2 - 2p + 1) + (3p^2 - 5p + 3) \]
Combine like terms:
- \( p^2 + 3p^2 = 4p^2 \)
- \( -2p - 5p = -7p \)
- \( 1 + 3 = 4 \)

So, we have:
\[ A + B = 4p^2 - 7p + 4 \]

5. **Substitute back into the perimeter equation**:
\[ 6p^2 - 4p + 9 = (4p^2 - 7p + 4) + C \]

6. **Isolate C**:
Rearranging gives:
\[ C = (6p^2 - 4p + 9) - (4p^2 - 7p + 4) \]

7. **Simplify the expression for C**:
Distributing the negative sign:
\[ C = 6p^2 - 4p + 9 - 4p^2 + 7p - 4 \]
Combine like terms:
- \( 6p^2 - 4p^2 = 2p^2 \)
- \( -4p + 7p = 3p \)
- \( 9 - 4 = 5 \)

Thus, we find:
\[ C = 2p^2 + 3p + 5 \]

### Final Answer:
The third side of the triangle is \( C = 2p^2 + 3p + 5 \).

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