The perimeter of a triangle is ` 5 - 3a + 7a^(2)` and two of its sides are `2a^(2) + 3a - 2 " and " 3a^(2) - a +3`.
Find the third side of the triangle.

Options

A. `4 - 5a + 2a^(2)`
B. `-4 + 5a - 2a^(2)`
C. `4 + 5a + 2a^(2)`
D. `5 - 4a - 2a^(2)`

Answer

To find the third side of the triangle, we can use the information given about the perimeter and the two sides.

### Step-by-Step Solution:

1. **Identify the perimeter and the sides**:
- The perimeter of the triangle is given as:
\[ P = 5 - 3a + 7a^2 \]
- The two sides of the triangle are:
\[ S_1 = 2a^2 + 3a - 2 \]
\[ S_2 = 3a^2 - a + 3 \]

2. **Let the third side be \( S_3 \)**:
- We can express the perimeter in terms of the sides:
\[ P = S_1 + S_2 + S_3 \]

3. **Substitute the known values into the perimeter equation**:
- Substitute \( S_1 \) and \( S_2 \) into the perimeter equation:
\[ 5 - 3a + 7a^2 = (2a^2 + 3a - 2) + (3a^2 - a + 3) + S_3 \]

4. **Combine the sides \( S_1 \) and \( S_2 \)**:
- Combine like terms:
\[ S_1 + S_2 = (2a^2 + 3a - 2) + (3a^2 - a + 3) = (2a^2 + 3a^2) + (3a - a) + (-2 + 3) \]
\[ = 5a^2 + 2a + 1 \]

5. **Set up the equation for \( S_3 \)**:
- Now we can rewrite the equation:
\[ 5 - 3a + 7a^2 = (5a^2 + 2a + 1) + S_3 \]

6. **Isolate \( S_3 \)**:
- Rearranging gives:
\[ S_3 = (5 - 3a + 7a^2) - (5a^2 + 2a + 1) \]

7. **Simplify the expression for \( S_3 \)**:
- Combine like terms:
\[ S_3 = (7a^2 - 5a^2) + (-3a - 2a) + (5 - 1) \]
\[ = 2a^2 - 5a + 4 \]

### Final Answer:
The third side of the triangle is:
\[ S_3 = 2a^2 - 5a + 4 \]