Question 42162
You cannot use synthetic division directly, but if you factor the denominator into  (t + 3)(t + 4), you can do it twice, once for each divisor...
This interface makes doing synthetic division awkward, but it goes like this...
-3_____1_____10_____35_____50_____24
_____________-3____-21____-42____-24
_____________________________________
_______1______7_____14______8______0

-4_____1______7_____14______8
_____________-4____-12_____-8
_____________________________
_______1______3______2

x^2 + 3x + 2 is the answer