The perimeter of a triangle is `8+13a+7a^(2)` and two of its sides are `2a^(2)+3a+2` and `3a^(2)-4a-1`. Find the third side of the triangle.

Answer

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

### Step 1: Understand the given information
We know that:
- The perimeter of the triangle is \( P = 8 + 13a + 7a^2 \).
- The first side of the triangle is \( S_1 = 2a^2 + 3a + 2 \).
- The second side of the triangle is \( S_2 = 3a^2 - 4a - 1 \).

### Step 2: Let the third side be \( S_3 \)
Let the third side of the triangle be \( S_3 = x \).

### Step 3: Write the equation for the perimeter
The perimeter of a triangle is the sum of all its sides:
\[ P = S_1 + S_2 + S_3 \]
Substituting the values we have:
\[ 8 + 13a + 7a^2 = (2a^2 + 3a + 2) + (3a^2 - 4a - 1) + x \]

### Step 4: Simplify the equation
Combine the two sides:
\[ 8 + 13a + 7a^2 = (2a^2 + 3a + 2 + 3a^2 - 4a - 1) + x \]
Combine like terms on the right side:
\[ 8 + 13a + 7a^2 = (2a^2 + 3a^2) + (3a - 4a) + (2 - 1) + x \]
This simplifies to:
\[ 8 + 13a + 7a^2 = 5a^2 - a + 1 + x \]

### Step 5: Solve for \( x \)
Now, we can isolate \( x \):
\[ x = (8 + 13a + 7a^2) - (5a^2 - a + 1) \]
Distributing the negative sign:
\[ x = 8 + 13a + 7a^2 - 5a^2 + a - 1 \]
Combine like terms:
\[ x = (7a^2 - 5a^2) + (13a + a) + (8 - 1) \]
This simplifies to:
\[ x = 2a^2 + 14a + 7 \]

### Step 6: Conclusion
Thus, the third side of the triangle is:
\[ \boxed{2a^2 + 14a + 7} \]