The sides of a triangle are `x^(2) - 3xy + 8, 4x^(2) + 5xy - 3` and `6 - 3x^(2) + 4xy`. Find its perimeter.

Answer

To find the perimeter of the triangle with sides given by the expressions \(x^2 - 3xy + 8\), \(4x^2 + 5xy - 3\), and \(6 - 3x^2 + 4xy\), we need to sum these three expressions.

### Step-by-step Solution:

1. **Identify the sides of the triangle:**
- Side 1: \(a = x^2 - 3xy + 8\)
- Side 2: \(b = 4x^2 + 5xy - 3\)
- Side 3: \(c = 6 - 3x^2 + 4xy\)

2. **Write the expression for the perimeter:**
The perimeter \(P\) of the triangle is given by the sum of its sides:
\[ P = a + b + c \]

3. **Substitute the expressions for the sides:**
\[ P = (x^2 - 3xy + 8) + (4x^2 + 5xy - 3) + (6 - 3x^2 + 4xy) \]

4. **Combine like terms:**
- Combine the \(x^2\) terms:
\[ x^2 + 4x^2 - 3x^2 = (1 + 4 - 3)x^2 = 2x^2 \]
- Combine the \(xy\) terms:
\[ -3xy + 5xy + 4xy = (-3 + 5 + 4)xy = 6xy \]
- Combine the constant terms:
\[ 8 - 3 + 6 = 8 - 3 + 6 = 11 \]

5. **Write the final expression for the perimeter:**
Putting it all together, we get:
\[ P = 2x^2 + 6xy + 11 \]

### Final Answer:
The perimeter of the triangle is:
\[ P = 2x^2 + 6xy + 11 \]