1. **State the problem:** Given the equation $$\frac{12x + 178}{12 + x} = \frac{A}{x} + \frac{B}{x - 1}$$, find the value of $B$. 2. **Understand the problem:** This is a partial fraction decomposition problem where the left side is expressed as a sum of fractions with denominators $x$ and $x-1$. 3. **Multiply both sides by the common denominator:** The common denominator is $x( x - 1)(12 + x)$. Multiply both sides by $12 + x$ to clear the denominator on the left: $$12x + 178 = A \frac{x( x - 1)(12 + x)}{x} + B \frac{x( x - 1)(12 + x)}{x - 1}$$ Simplify the fractions: $$12x + 178 = A (x - 1)(12 + x) + B x (12 + x)$$ 4. **Substitute $x = 1$ to find $B$:** Substituting $x = 1$ cancels the term with $A$ because $(x - 1)$ becomes zero: $$12(1) + 178 = A (1 - 1)(12 + 1) + B (1)(12 + 1)$$ $$12 + 178 = 0 + B \times 13$$ $$190 = 13B$$ 5. **Solve for $B$:** $$B = \frac{190}{13}$$ 6. **Final answer:** $$\boxed{\frac{190}{13}}$$