Dividing polynomial $f(x)$ by $(x-a)(x-b)(x-c)$ 
$f(x)$ is a polynomial with a degree greater than 3. When $f(x)$ is divided by $(x-a)(x-b)(x-c)$, prove remainder is
  $$\frac{f(a)(x-b)(x-c)}{(a-b)(a-c)}+ \frac{f(b)(x-a)(x-c)}{(b-c)(b-a)} +\frac{f(c)(x-a)(x-b)}{(c-b)(c-a)}$$

My Try
I tried this using the conventional method,$$f(x)=Q(x)(x-a)(x-b)(x-c)+Ax^2+Bx+C$$
But then I got long answers for coefficients $A$, $B$ & $C$.
Is there a better way to solve this? Can anyone give me a hint to work this?

We have:
$$f(x)=Q(x)(x-a)(x-b)(x-c)+Ax^2+Bx+C$$
as you mentioned. Now note that 
$$f(a)=Aa^2+Ba+C\\
f(b)=Ab^2+Bb+C\\
f(c)=Ac^2+Bc+C$$
Now it is easy to solve this system of equations for $A$, $B$ and $C$.
