<s>[INST] If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist? 
If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist?

*

*One

*Two

*Three

*Infinitely many


Also, what are these pairs?
 [/INST] Sum of the roots is $a+b = -a \implies 2a+b = 0$.
Product of the roots is $ab = b \implies b = 0 \text{ or } a = 1$.
If $b=0$, then $2a+ 0 = 0 \implies a = 0$.
If $a=1$, then $2+b = 0 \implies b = -2$.
</s>