<s>[INST] I need to define the cubic equation $f(x)= ax^3 + bx^2 + cx + d$ It is given that if cubic function $f(x)$ is divided by $(x^2+4)$ and $(x+3)$, the remainders will be $7x-20$ and $-2$ respectively. Also it is said show that $f(x)$ is $x^3 + 6x^2 + 11x +4$. I tried but can i find the $a,b,c$, and $d$ without knowing them in from the start?
 [/INST] Solving in pari/gp:
? chinese(Mod(7*x-20,x^2+4),Mod(-2,x+3))
%1 = Mod(3*x^2 + 7*x - 8, x^3 + 3*x^2 + 4*x + 12)
?
? Mod(a*x^3+b*x^2+c*x+d,(x^2+4)*(x+3))
%2 = Mod((-3*a + b)*x^2 + (-4*a + c)*x + (-12*a + d), x^3 + 3*x^2 + 4*x + 12)

I.e. $3=-3a+b$, $7=-4a+c$, $-8=-12a+d$.
And $\quad a=1\quad\Longrightarrow\quad b=6,c=11,d=4$.
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