<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] Since $a, b$ are zeros, they each satisfy the equation:
$$x=a: \quad a^2 + a\cdot a + b = 0\iff 2a^2 + b = 0 \iff b = -2a^2$$
$$x = b:\quad b^2 + ab + b = 0$$
Substituting $b = -2a^2$ into the second equation: $$(-2a^2)^2 + a(-2a^2) + -2a^2 = 0 \iff 4a^4 -2a^3 - 2a^2 = 2a^2(2a^2 -a - 1) = 0$$
The right-hand factor itself factors nicely, giving you 3 solutions to $a$, and then you can find the corresponding value for $b$ by using $b = -2a^2$.
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