Let $f(x)$ be a polynomial function such that $f(x)+f'(x)+f''(x)=x^5+64$. Then, the value of $\lim\_{x \to 1}\frac{f(x)}{x-1}$ is

$\boxed{A) \; -15}$

$B) \; -60$

$C) \; 60$

$D) \; 15$

I have solved it by considering a $5$ degree general polynomial of the form $ax^5+bx^4+cx^3+dx^2+ex+f$ and then algebraically solving to obtain the required result.

Is there any other method which doesn't include assumption of a function of any kind?

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Source: JEE Mains 2022 25th June Shift-1