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 

$$  \left(   x^{2}  +  x  + 1 \right)  \left(   \frac{  -  x  - 7 }{ 31 }  \right)  -  \left(   x^{2}  + 2 x  - 4 \right)  \left(   \frac{  -  x  - 6 }{ 31 }  \right)  =  \left( -1  \right)  $$
is all you need. Cleaning up,
$$  (x+7) \left(   x^{2}  +  x  + 1 \right) - (x+6)  \left(   x^{2}  + 2 x  - 4 \right) = 31 $$
There is no guarantee of integer coefficients, even when both polynomials have integer coefficients. The units in $\mathbb Q[x]$ are nonzero rational constants.
