Choice C is correct. It's given that the function $f$ is defined by $f (x) = |x - 4 x|$ . It's also given that $f (5) - f (a) = - 15$ . Substituting $5$ for $x$ in the function $f (x) = |x - 4 x|$ yields $f (5) = |5 - 4 (5)|$ and substituting $a$ for $x$ in the function $f (x) = |x - 4 x|$ yields $f (a) = |a - 4 a|$ . Therefore, $f left parenthesis 5 right parenthesis equals 15$ and $f (a) = |- 3 a|$ . Substituting $15$ for $f left parenthesis 5 right parenthesis$ and $StartAbsoluteValue minus 3 a EndAbsoluteValue$ for $f (a)$ in the equation $f (5) - f (a) = - 15$ yields $15 - |- 3 a| = - 15$ . Subtracting $15$ from both sides of this equation yields $- |- 3 a| = - 30$ . Dividing both sides of this equation by $negative 1$ yields $|- 3 a| = 30$ . By the definition of absolute value, if $|- 3 a| = 30$ , then $- 3 a = 30$ or $- 3 a = - 30$ . Dividing both sides of each of these equations by $negative 3$ yields $a = - 10$ or $a equals 10$ , respectively. Thus, of the given choices, a value of $a$ that satisfies $f (5) - f (a) = - 15$ is $10$ .

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

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

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