Strange wording of the problem -- it asks to FIND one factor; but it already SHOWS one factor....
(1) Show that x+3 is a factor using synthetic division:
-3 | 8 1 -55 42
|
+-------------------
-3 | 8 1 -55 42
| -24
+-------------------
8 -23
-3 | 8 1 -55 42
| -24 69
+-------------------
8 -23 14
-3 | 8 1 -55 42
| -24 69 -42
+-------------------
8 -23 14 0
(x+3) is a root; and
There are many methods for factoring that quadratic; I am old school and think the best thing to do is try to use logical reasoning to find the pair of linear factors.
With this one, that leads quickly to the correct factorization.
We have two choices for the form of the factorization:
(4x-?)(2x-)
(8x-?)(x-?)
But logical reasoning quickly tells me the first is impossible, because the middle term of the product is going to be even.
So the factorization is of the form (8x-?)(x-?); quick trial with the factors of 14 leads us to (8x-7)(x-2).
The complete factorization is
The roots are -3, 7/8, and 2.