Explanation:
any polynomial is factored by terms of its zeros (the terms of the x- values creating a 0-y-value as result).
y = (x - a)(x - b)(x - c)...
means that for x = a or b or c or ..., y = 0
and (x - a), (x - b), (x - c), ... are all the factors of the y-expression
so,
2x + 1 = 0
2x = -1
x = -1/2
so, for x = -1/2 our term (2x + 1) = 0.
if now the whole expression is 0 for x = -1/2, then (2x + 1) is a factor :
6×(-1/2)³ + 13×(-1/2)² + 17×(-1/2) + 6 =
= -6×1/8 + 13×1/4 - 17×/1/2 + 6 =
= -6/8 + 13/4 - 17/2 + 6 = -3/4 + 13/4 - 17/2 + 6 =
= 10/4 - 17/2 + 6 = 5/2 - 17/2 + 6 = -12/2 + 6 =
= -6 + 6 = 0
that is why (2x + 1) is indeed a factor of
6x³ + 13x² + 17x + 6