Answer: C. $2x^{2}+5x-42$

Explanation:1. Method 1: Expand the product of the binomials first. Consider the product $(x-\dfrac{7}{2})(x+6)$. Using the FOIL method (First, Outer, Inner, Last): First: $x \times x = x^2$ Outer: $x \times 6 = 6x$ Inner: $-\dfrac{7}{2} \times x = -\dfrac{7}{2}x$ Last: $-\dfrac{7}{2} \times 6 = -\dfrac{42}{2} = -21$ Combine these terms: $x^2 + 6x - \dfrac{7}{2}x - 21$.

2. Combine the x terms: $6x - \dfrac{7}{2}x = \dfrac{12}{2}x - \dfrac{7}{2}x = \dfrac{12-7}{2}x = \dfrac{5}{2}x$. So, $(x-\dfrac{7}{2})(x+6) = x^2 + \dfrac{5}{2}x - 21$.

3. Multiply the result by $2$: The original expression is $2 \times (x^2 + \dfrac{5}{2}x - 21)$. Distribute the $2$ to each term inside the parenthesis: $2(x^2) + 2(\dfrac{5}{2}x) - 2(21)$ $2x^2 + \dfrac{10}{2}x - 42$ $2x^2 + 5x - 42$

4. Method 2: Distribute the $2$ into one of the binomials first. Let's distribute $2$ into $(x-\dfrac{7}{2})$: $2(x-\dfrac{7}{2}) = 2x - 2(\dfrac{7}{2}) = 2x - 7$. Now multiply $(2x-7)(x+6)$. Using FOIL: First: $2x \times x = 2x^2$ Outer: $2x \times 6 = 12x$ Inner: $-7 \times x = -7x$ Last: $-7 \times 6 = -42$ Combine these terms: $2x^2 + 12x - 7x - 42 = 2x^2 + 5x - 42$.

5. Check the solution with the options: The equivalent expression is $2x^{2}+5x-42$. This matches option [C].

6. Conclusion: The expression equivalent to $2(x-\dfrac{7}{2})(x+6)$ is $2x^{2}+5x-42$.