Problem about the number of values the sum of two variables can take.

Question

Given that $a, b, c$ and $d%$ are real numbers, such that

how many values can $a+b$ take?

I know we can make four equations from this:

$a+ab=c$

$a+1=d$

$a+b^2 = -d$

$b+1 = -c$

and these can be reduced to

$a+ab = -b-1$

$a+1 = -a-b^2$

but I don't know how to go on from here as these are not linear equations. Thanks in advance for any help.

Answer 1

$a+ab=-b-1$

$\Leftrightarrow a(b+1)+b+1=0$

$\Leftrightarrow (a+1)(b+1)=0$

We can easily solve $a,b$ from here and solve the second equation for two cases: $a=-1$ and $b=-1.$