A practical method is as follow:
-2x^3+2x+3 &&&&& x^3\\
so the quotient is $x^4+x^3-2x^2-4x-6$ and the remainder is $-9$.
and to explain the procedure of calculus: we divide the leading term $x^5$ of the dividend by the leading term $x$ of the divisor we find $x^4$ and then we calculate:
and repeat the same calculus using $R(x)$ as your new dividend until we find the remainder $R(x)$ with degree less than the degree of the divisor $x-2$.