After that, however, it subtracted (dividend/2^31) from the result.
Actually, it subtracts dividend >> 31, which is -1 for negative dividend, and 0 for non-negative dividend, when right-shifting negative integers is arithmetic right-shift (and int is 32 bits wide).
0x6666667 = (2^34 + 6)/10
So for x < 0, we have, writing x = 10*k + r with -10 < r <= 0,
0x66666667 * (10*k+r) = (2^34+6)*k + (2^34 + 6)*r/10 = 2^34*k + 6*k + (2^34+6)*r/10
Now, arithmetic right shift of negative integers yields the floor of v / 2^n, so
(0x66666667 * x) >> 34
results in
k + floor((6*k + (2^34+6)*r/10) / 2^34)
So we need to see that
-2^34 < 6*k + (2^34+6)*r/10 < 0
The right inequality is easy, both k and r are non-positive, and not both are 0.
For the left inequality, a bit more analysis is needed.
r >= -9
so the absolute value of (2^34+6)*r/10 is at most 2^34+6 - (2^34+6)/10.
|k| <= 2^31/10,
so |6*k| <= 3*2^31/5.
And it remains to verify that
6 + 3*2^31/5 < (2^34+6)/10
1288490194 < 1717986919
Yup, true.