The document you provided contains several mathematical tasks, primarily in algebra and matrix operations. Here’s an overview of the questions in each task:

1. Problem 1: Complex numbers.

   • $a) \; \left(\sqrt{3} + i\right)^{15}$
   • $b) \; \sqrt{1 - \sqrt{3}i}$

2. Problem 2: Polynomial division and canonical forms.

   • Part A: Divide $f(x) = x^4 - 6x^3 + 16x^2 - 24x + 16$ by $g(x) = x^2 - 3x - 1$ using a specified method (likely the least-squares method).
   • Part B: Divide $f(x) = x^4 - 6x^3 + 16x^2 - 24x + 16$ by $g(x) = x + 2$ and evaluate $f(-2)$.
   • Part C and D: Write the canonical forms of $f(x)$ for the real and complex number sets, respectively.

3. Problem 3: Matrix operations.

   • Given matrix $A$, calculate:
     • Part A: The inverse of matrix $A$.
     • Part B: Solve the matrix equation $X \cdot A = B$, where both matrices $A$ and $B$ are provided.

4. Problem 4: Functions of matrices and solving matrix equations.

   • Part A: Apply the function $f(x) = x^2 - 3x + 2$ to matrix $A$.
   • Part B: Solve the matrix equation $B \cdot X = C$, where $B$ and $C$ are given matrices.

5. Problem 5: Determinant calculation.

   • Compute the determinant of a 5x5 matrix.

6. Problem 6: Solving systems of linear equations.

   • Part A: Solve a system of three equations with three unknowns.
   • Part B: Solve a system of four equations with four unknowns, with additional conditions on the parameter $x_2$.

Would you like to go through any specific problem in detail, or would you like the solutions for all parts sequentially?

Further Questions You Might Explore:

1. How to calculate powers of complex numbers using De Moivre's theorem?
2. What are the steps involved in polynomial division using synthetic division?
3. How to find the canonical form of a polynomial for real and complex domains?
4. Methods to find the inverse of a matrix and properties that ensure its existence.
5. Techniques for solving systems of linear equations with parameters.

Tip: For complex numbers, expressing them in polar form simplifies calculations, especially when raising them to powers.