The smallest integer that can be added to -2m3 − m + m2 + 1 to make it completely divisible by m + 1 is

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.

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