First, find out what the sum is. Consider 3 rows of 3 numbers: each row has sum S. The rows contain all the numbers, so since the sum of the numbers is -45, 3S=-45 and S=-15.
Take each number in turn and work out what other numbers can go with it to make -15:
0: (-10,-5),(-8,-7)
-2: (-10,-3),(-8,-5),(-7,-6)
-3: (-10,-2),(-8,-4),(-7,-5)
-4: (-9,-2),(-8,-3),(-6,-5)
-5: (-10,0),(-8,-2),(-7,-3),(-6,-4)
-6: (-7,-2),(-5,-4)
-7: (-8,0),(-6,-2),(-5,-3)
-8: (-7,0),(-5,-2),(-4,-3)
-10: (-5,0),(-3,-2)
Now, consider the square:
C1 C2 C3
C4 C5 C6
C7 C8 C9
Each cell participates in various sums. C1, for example, is in the first row and column and on a diagonal, so it is involved in 3 sums. C3, C7 and C9 are also similarly involved. Here's a complete list:
C2,C4,C6,C8: 2 sums
C1,C3,C7,C9: 3 sums
C5: 4 sums
Now we match the number of sums with the numbers that can be involved in at least the same number of sums.
C5=-5.
2 sums: {C2 C4 C6 C8}={0 -6 -10} (exact) < {-2 -3 -4 -7 -8} (3 sums)
3 sums: {C1 C3 C7 C9}={-2 -3 -4 -7 -8}, one of these can be used to satisfy 2 sums, leaving 4 to match the cells.
So {C1 C3 C7 C9}={-2 -3 -4 -7)|{-2 -3 -4 -8}|{-2 -3 -7 -8}|-2 -4 -7 -8}|{-3 -4 -7 -8} and
{C2 C4 C6 C8}={0 -6 -8 -10}|{0 -6 -7 -10}|{0 -4 -6 -10}|{0 -3 -6 -10}|{0 -2 -6 -10}
Because of symmetry we can assign 0 to C2. So C8=-10:
C1 0 C3
C4 -5 C6
C7 -10 C9
C7+C9=-5, which means only -2 and -3. Because of symmetry we can assign either number to C7:
C1 0 C3
C4 -5 C6
-2 -10 -3
C1-8=-15 so C1=-7; C3-7=-15, so C3=-8:
-7 0 -8
C4 -5 C6
-2 -10 -3
C4=-6 and C6=-4:
-7 0 -8
-6 -5 -4
-2 -10 -3