The image contains mathematical tasks involving simplifying fractions and algebraic expressions. Let's focus on problem 4.1, which reads:

$\frac{16 - x}{4 + \sqrt{x}} \cdot \frac{x - 5 + \sqrt{x}}{(x - 3)(x - 1)}.$

Steps to Simplify:

1. Rewrite the numerator and denominator of both fractions clearly.

   • The first fraction: $\frac{16 - x}{4 + \sqrt{x}}$.
   • The second fraction: $\frac{x - 5 + \sqrt{x}}{(x - 3)(x - 1)}$.

2. Combine the fractions by multiplying the numerators and denominators. $\frac{(16 - x)(x - 5 + \sqrt{x})}{(4 + \sqrt{x})(x - 3)(x - 1)}.$

3. Factorize wherever possible:

   • Notice $16 - x = (4 - \sqrt{x})(4 + \sqrt{x})$ (difference of squares).

   Substituting this, the numerator becomes: $(4 - \sqrt{x})(4 + \sqrt{x})(x - 5 + \sqrt{x}).$

4. Cancel common terms:

   • The $4 + \sqrt{x}$ in the numerator and denominator cancels out.

   The simplified expression is: $\frac{(4 - \sqrt{x})(x - 5 + \sqrt{x})}{(x - 3)(x - 1)}.$

Would you like me to expand this further, or solve another problem from the image?

Relative Questions:

1. How do you handle complex fractions with square roots effectively?
2. What does factoring the difference of squares involve in algebraic expressions?
3. Why is simplifying algebraic expressions crucial in solving equations?
4. How does multiplying and dividing fractions work with polynomials?
5. What are common mistakes to avoid when simplifying expressions?

Tip: Always check for opportunities to cancel terms after factoring—it simplifies your work significantly!