Choice B is correct. The solutions to a quadratic equation of the form $a x^{2} + b x + c = 0$ , where $a$ , $b$ , and $c$ are constants, can be calculated using the quadratic formula $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ . The given equation is of this form, where $a equals 4$ , $b = - 5$ , and $c = - 5$ . Substituting these values into the quadratic formula yields $x = \frac{- (- 5) \pm \sqrt{{(- 5)}^{2} - 4 (4) (- 5)}}{2 (4)}$ , or $x = \frac{5 \pm \sqrt{25 + 80}}{8}$ . These solutions can be rewritten as $x = \frac{5 \pm \sqrt{105}}{8}$ , or $x = \frac{5}{8} \pm \frac{\sqrt{105}}{8}$ . Since $\frac{5}{8} + \frac{\sqrt{105}}{8}$ is greater than $\frac{5}{8} - \frac{\sqrt{105}}{8}$ , it follows that the greatest solution to the given equation is $\frac{5}{8} + \frac{\sqrt{105}}{8}$ .

Choice A is incorrect. This is the least, not the greatest, solution to the given equation.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

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