Find the quotient and remainder Find the quotient and remainder when $x^6+x^3+1$ is divided by $x+1$
Let $f(x)=x^6+x^3+1$
Now $f(x)=(x+1).q(x) +R $  where r is remainder 
Now putting $x=-1$ we get $R=f(-1)$
i.e $R=1-1+1=1$
Now $q(x)=(x^6+x^3)/(x+1)$
But what I want to know if there is another way to get the quotient except simple division. 

You're on the right track:
$$
x^6+x^3=x^3(x^3+1)=x^3(x+1)(x^2-x+1)
$$
Therefore
$$
q(x)=(x^6+x^3)/(x+1)=x^3(x^2-x+1)=x^5-x^4+x^3
$$
