1. Find all pairs of integers $a, b$ such that the sum $a+b$ is a root of the equation $x^{2}+a x+b=0$.
1. We are looking for integers $a, b$ such that $(a+b)^{2}+a(a+b)+b=0$, which is a quadratic equation in the unknown $b$: $b^{2}+(3 a+1) b+2 a^{2}=0$ with integer coefficients. This equation has an integer root only if its discriminant

$$
D=(3 a+1)^{2}-4 \cdot 2 a^{2}=(a+3)^{2}-8
$$

is a perfect square. This perfect square is 8 less than another perfect square $(a+3)^{2}$. As we can easily determine (the differences between the squares of two consecutive natural numbers increase), the only perfect squares that differ by 8 are 9 and 1, so $(a+3)^{2}=9$, from which we get $a=-6$ or $a=0$. For $a=-6$, we get $b=8$ and $b=9$, and for $a=0$, we get $b=0$ and $b=-1$. Thus, we obtain four solutions: $(a, b)$ is one of the pairs $(-6,8),(-6,9),(0,0),(0,-1)$.

Notes. If we choose $a$ as the unknown instead of $b$, we get the equation $2 a^{2}+3 b a+\left(b^{2}+b\right)=0$ with the discriminant $D^{\prime}=9 b^{2}-8 \cdot\left(b^{2}+b\right)=(b-4)^{2}-16$; the perfect squares differing by 16 are only 0,16 and 9,25.

The problem of finding two perfect squares $x^{2}$ and $y^{2}$ with a given difference $d$ can be solved for small values of $d$ (such as $d=8$ or $d=16$ in our problem) by testing the first few squares 0,1, $4,9, \ldots$ For any natural number $d$, we can proceed by rewriting the equation $x^{2}-y^{2}=d$ as $(x-y)(x+y)=d$ and listing all factorizations of the given number $d$ into the product $d_{1} d_{2}$ of two integer factors; from the equations $d_{1}=x-y, d_{2}=x+y$ we then compute the corresponding $x$ and $y$.

For a complete solution, award 6 points. Consider a solution complete even if the perfect squares differing by 8 or 16 are listed (guessed) without explanation of why no other such squares exist. For finding solutions only for $a=0$ (e.g., by the incorrect reasoning that the number $a+b$ divides the number $b$ only for $a=0$), award 1 point. For guessing all four solutions (e.g., by experimenting with small numbers $a, b$), award 2 points.