hello, for the variable $a$ i have found the equation $a^3+3a^2+3a-1=0$. Sonnhard.

WakeUp wrote: Determine all real numbers $a,b,c,d$ that satisfy the following equations \[\begin{cases} abc + ab + bc + ca + a + b + c = 1\\ bcd + bc + cd + db + b + c + d = 9\\ cda + cd + da + ac + c + d + a = 9\\ dab + da + ab + bd + d + a + b = 9\end{cases}\] Put $x:=a+1$ and $ y,z,u$ similarly. Then $xyz=2, yzu=zux=uxy=10$. Thus $x=y=z$ and $a=b=c=\sqrt[3]{2}-1,d=\sqrt[3]{250}-1$.

add 1 at each eq and factorize

2nd - 3rd---->$\boxed{b=a}$ 3rd- 4th--->$\boxed{c=b}$ 1st--->$(a-1)^3=0$---->$a=1$ put a to 4th and we are done