Answer: B. $-50$

Explanation:1. Understand the condition for exactly one solution in a quadratic equation: A quadratic equation $ax^2 + bx + c = 0$ has exactly one solution (or a repeated real root) if and only if its discriminant is equal to zero. The discriminant is given by the formula: $\Delta = b^2 - 4ac$. So, for exactly one solution, $b^2 - 4ac = 0$.

2. Identify the coefficients a, b, and the constant (which is 'c' in the discriminant formula, but represented by a different 'c' in the problem's equation): From the given equation $-2x^2 + 20x + c = 0$: $a = -2$ $b = 20$ The constant term in the equation is 'c' (let's call it $c\_{eq}$ to avoid confusion with the discriminant's 'c'). So in the discriminant formula $b^2-4ac\_{gen}=0$, $c\_{gen}$ corresponds to $c\_{eq}$ from the problem.

3. Set the discriminant to zero and solve for c (which is $c\_{eq}$): $(20)^2 - 4(-2)(c) = 0$ $400 - (-8)(c) = 0$ $400 + 8c = 0$ $8c = -400$ $c = \dfrac{-400}{8}$ $c = -50$

4. Compare with the given options: The calculated value $c = -50$ matches option [B].

Conclusion:The value of c for which the equation has exactly one solution is $-50$.