Write the given polynomial as
f(m) = -2m³ + m² - m + 1
If f(m) is to be completely divisible by (m+1), then (m+1) is a factor, and m = -1 is a zero of f(m).
Let k = the integer to be added to f(m), so that the new polynomial is
-2m³ + m² - m + k + 1
Perform synthetic division.
-1 | -2 1 -1 (k+1)
2 -3 4
-----------------------
-2 3 -4 k+5
To make the remainder equal to zero means that
k + 5 = 0 => k = -5.
Answer: The integer to add is -5.
Note: If you do not know synthetic division, use long division
- 2m² + 3m - 4
----------------------------------
m+1 | -2m³ + m² - m + (k+1)
-2m³ - 2m²
----------------------------------
3m² - m + (k+1)
3m² + 3m
-----------------------
- 4m + (k+1)
- 4m - 4
----------------
k+5
To make the remainder (k+5) = 0, make k = -5