9514 1404 393
Answer:
3
Explanation:
What you're looking for is the remainder from the division of the cubic by the given divisor (2x+1). The number to be added will be the opposite of that remainder.
There are several ways you can find the remainder from the division. Shown in the attachment is polynomial long division. That shows the remainder is -3, so the number to be added is +3.
Another way to find the remainder is to do synthetic division by (2x+1). This is equivalent to synthetic division by x+1/2, so the number used in the tableau is -1/2. This, too, will show a remainder of -3.
Finally, the remainder can be found by evaluating the cubic using the value of x that is the root of the divisor: -1/2.
((4x -10)x +12)x +6 for x=-1/2 is ...
((4(-1/2) -10)(-1/2) +12)(-1/2) +6 = 18(-1/2) +6 = -3
So, 3 should be added to make the cubic exactly divisible by 2x+1.