How to use Chinese Remainder Theorem A cubic polynomial $f(x)=ax^3+bx^2+cx+d$ gives remainders $-3x+3$ and $4x-1$ when divided by $x^2+x+1$ and $x^2+2x-4$. Find the value of $a,b,c,d$.
I know it’s easy but i wanna use Chinese Remainder Theorem(and Euclidean Algorithm) to solve it. 
A hint or a detailed answer would be much appreciated 

EDIT: I just noticed my answer below doesn't meet OP's expectations, as I'm not explicitely using the Chinese theorem. I haven't read the question carefully enough...  

Reduction of $f(x)$ modulo $x^2+x+1$ can be computed by successively replacing all instances of $x^2$ in $f$ by $-x-1$, until we obtain an expression of degree $<2$. The process reduces $f(x)$ to $a(-x-1)x +b(-x-1) +cx +d=-ax^2+(c-a-b)x+d-b$, which in turns gives $ -a(-x-1)+(c-a-b)x+d-b = (c-b)x+a+d-b$.  
The same method with $x^2+2x-4$ tells us that $f(x)$ reduces to $(c-2b+8a)x-8a+d$. 
These two reductions must be respectively $-3x+3$ and $4x-1$. By very definition of equality between polynomials (that is, we can identify the coefficients), it reduces to the following system with $4$ equations and $4$ unknowns, which I let your solve by yourself:
$$\left\{
\begin{array}{}
-3&=c-b\\
3&=a+d-b\\
4&=c-2b+8a\\
-1&=-8a+d\\
\end{array}
\right.$$
