Explanation

(a)
Given, $P(x) = x + ax + b$
$\therefore\ \ \ \ \ \ \ \ \ P(10) = 10 + 10a + b = 10 + 5 = 15$
And $P(11) = 11 + 11a + b$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 11 + 5 + a = 16 + a$
$\because\ \ \ \ \ P(10)P(11) = P(n)$
$\Rightarrow \ \ \ \ \ 15(16 + a) = n + na + b$
$\Rightarrow \ \ \ \ \ \ 240 + 15a = n + na + 5 - 10a$
$\Rightarrow \ \ \ \ n + na - 25a - 235 = 0$
(a) When $n = 15$
$15 + 15a - 25a - 235 = 0$
$\Rightarrow \ \ \ \ \ \ a = - 22$ and $b = 225$
(b) When $n = 64$
$65 + 65a - 25a - 235 = 0$
$\Rightarrow \ \ \ \ \ \ a = - \frac{17}{4}$ which is not integer.
(c) When $n = 115$
$\ \ \ \ 115 + 115a - 25a - 235 = 0$
$\Rightarrow \ \ \ \ \ \ \ \ a = \frac{4}{3}$ which is not integer.
(d) When $n = 165$
$\ \ \ \ \ 165 + 165a - 25a - 235 = 0$
$\Rightarrow \ \ \ \ a = \frac{1}{2}$ which is not integer.