To solve the provided equations, we can take them one by one.

1. **Equation:**
\(\frac{1 - x}{x + 1} + 3 = 2x + 3\)

2. **Equation:**
\(\frac{2x + 1}{x + 1} + \frac{2}{x} = \frac{2}{x(x + 1)}\)

3. **Equation:**
\(\frac{x - 3}{x - 2} + \frac{x - 2}{x - 4} = -1\)

4. **Equation:**
\(\frac{x}{x + 6} + \frac{3}{x - 8} = \frac{-12x + 33}{(x + 6)(x - 8)}\)

5. **Equation:**
\(\frac{2x + 1}{x - 1} + \frac{2x - 1}{x^2 - 1} = \frac{x^2 - 3x + 2}{x^2 - 1}\)

### Steps to Solve Each Equation:

1. **First Equation:**
Cross-multiply to find a common denominator and rearrange terms to isolate \(x\).

2. **Second Equation:**
Combine fractions on the left side to consolidate and set it equal to the right side.

3. **Third Equation:**
Cross-multiply to eliminate fractions and solve for \(x.\)

4. **Fourth Equation:**
Combine terms under a common denominator and simplify the equation.

5. **Fifth Equation:**
Notice that \(x^2 - 1\) can be factored and cancelled out if not equal to zero. Rearranging will lead to solving for \(x\).

For detailed solving, please let me know which specific equation you want to focus on!