Answer: C. $\dfrac{61c}{4} + 105$

Explanation:1. Evaluate $g(c+7)$ using the definition of $g(x)$: Given $g(x) = b - 15x$. Substitute $x = c+7$ into the function: $g(c+7) = b - 15(c+7)$ Distribute the -15: $g(c+7) = b - 15c - 15 \times 7$ $g(c+7) = b - 15c - 105$

2. Use the given information $g(c+7)=\dfrac{c}{4}$: We have two expressions for $g(c+7)$. Set them equal to each other: $b - 15c - 105 = \dfrac{c}{4}$

3. Solve for b: To isolate b, add $15c$ and $105$ to both sides of the equation: $b = \dfrac{c}{4} + 15c + 105$

4. Combine the terms involving c: To add $\dfrac{c}{4}$ and $15c$, find a common denominator, which is 4. $15c = \dfrac{15c \times 4}{4} = \dfrac{60c}{4}$ So, $\dfrac{c}{4} + 15c = \dfrac{c}{4} + \dfrac{60c}{4} = \dfrac{c + 60c}{4} = \dfrac{61c}{4}$

5. Write the final expression for b: $b = \dfrac{61c}{4} + 105$

6. Compare with the given options: [A] $\dfrac{15c}{4} + 105$. Incorrect. [B] $\dfrac{c}{4} - 15c + 105 = \dfrac{c-60c}{4} + 105 = \dfrac{-59c}{4} + 105$. Incorrect. [C] $\dfrac{61c}{4} + 105$. This matches our derived expression. [D] $15c - \dfrac{c}{4} + 105 = \dfrac{60c-c}{4} + 105 = \dfrac{59c}{4} + 105$. Incorrect.

Conclusion:The expression that represents the value of b is $\dfrac{61c}{4} + 105$.