Simplify the following expression: $$\frac{b^2 - 1}{3b^2 - 4b + 1} \cdot \frac{3b - 1}{b} + \frac{1}{b}$$

Simplifying the expression:

Step-by-step solution:

1. Step 1: Factor the quadratic expression in the denominator of the first fraction.
   The expression is \(3b^2 - 4b + 1\). We look for two numbers that multiply to \(3 \cdot 1 = 3\) and add to \(-4\). These numbers are \(-3\) and \(-1\). So we can rewrite the expression as \(3b^2 - 3b - b + 1\).
   Factor by grouping: \(3b(b - 1) - 1(b - 1) = (3b - 1)(b - 1)\).
2. Step 2: Rewrite the first fraction with the factored denominator.
   The first fraction becomes \(\frac{b^2 - 1}{(3b - 1)(b - 1)}\).
3. Step 3: Factor the numerator of the first fraction.
   The numerator is a difference of squares: \(b^2 - 1 = (b - 1)(b + 1)\).
4. Step 4: Substitute the factored numerator and denominator back into the first fraction.
   The first fraction is now \(\frac{(b - 1)(b + 1)}{(3b - 1)(b - 1)}\).
5. Step 5: Simplify the first fraction by canceling out the common factor \((b - 1)\).
   The simplified first fraction is \(\frac{b + 1}{3b - 1}\).
6. Step 6: Perform the multiplication of the first term.
   Multiply the simplified first fraction by \(\frac{3b - 1}{b}\):
   \(\frac{b + 1}{3b - 1} \cdot \frac{3b - 1}{b}\).
   Cancel out the common factor \((3b - 1)\).
   The result of the multiplication is \(\frac{b + 1}{b}\).
7. Step 7: Add the second term to the result of the multiplication.
   The expression is now \(\frac{b + 1}{b} + \frac{1}{b}\).
8. Step 8: Combine the fractions since they have a common denominator.
   Add the numerators: \((b + 1) + 1 = b + 2\).
   The final simplified expression is \(\frac{b + 2}{b}\).

Answer: \(\frac{b + 2}{b}\)