Can you solve this system of equations with four variables?

[anemone]

Find all real solutions to the following system of equations:

$a+b+c+d=5$

$ab+bc+cd+da=4$

$abc+bcd+cda+dab=3$

$abcd=-1$

 

[anemone]

Solution that I saw somewhere online:

Spoiler

We're given

$a+b+c+d=5$

$ab+bc+cd+da=4$

$abc+bcd+cda+dab=3$

$abcd=-1$

Let $X=a+c$ and $Y=b+d$. Then the system of equations is equivalent to

$x+Y=5$

$XY=4$

$Xbd+Yac=3$

$(Xbd)(Yac)=-4$

The first two of these equations imply $(X,\,Y)=(1,\,4)$ and the last two give $(Xbd,\,Yac)=(4,\,-1)$.

And this yields:

$X$	$Y$	$Xbd$	$Yac$	$(a,\,c)$	$(b,\,d)$
1	4	4	-1	$\dfrac{1\pm 2}{2}$	2
1	4	-1	4	-	-
4	1	4	-1	-	-
4	1	-1	4	2	$\dfrac{1\pm 2}{2}$