I'm looking for all the complex solutions to the following 3 equations (and for this consider $a$ to be some given constant, so that there are really just 3 unknowns in solving):
$0=2abc+a^2+2ad+b+3d, \ \ \ 0 = a d^2 + b^2c + 2ab + 2bd + 3d^2, \ 0 = bd^2 + d^3 + b^2$
I find that all roots have $d = 0$ or $d = 9/2$ or $d = -a^2/(a+1)$. I also find that one root (at $d = -a^2/(a+1)$) has $cd = 1$ for all a. The two constant $d=9/2$ and $cd=1$ results are surprising, so was there some way for me to see this from the equations directly in hindsight? Better yet, is there a general method which shows results like this in foresight for similar equations? I'm guessing that a mathematician would look at these equations and some change of variables is obvious.
About your second question, a possible general algorithmic method to solve such problems is to compute a Gröbner basis for an elimination ideal.
For instance, by running the following code in the "magma calculator": http://magma.maths.usyd.edu.au/calc/ we can get automatically the three conditions:
K:=Rationals();
R$<$a,b,c,d$>$:=PolynomialRing(K,4);
I:=Ideal([2*a * b *c+a^2+2*a*d+b+3*d, a*d^2+b^2*c+2*a*b+2*b*d+3*d^2, b*d^2+d^3+b^2]);
J:=EliminationIdeal(I,{c,d});
f:=GroebnerBasis(J);
Factorization(f[1]);