Using synthetic division we can divide by the two divisors x-3 and x-2 and compare remainders:
3 | 1 a -3 4
1 3 3a+9 9a+18
1 a+3 3a+6 | 9a+22 remainder
2 | 1 a -3 4
1 2 2a+4 4a+2
1 a+2 2a+1 | 4a+6 remainder
The remainder from the first division is twice that of the second division so:
9a+22=2(4a+6), 9a+22=8a+12, a=-10.