Solution: f(x)=x? +2mx+ 1=(x+m)？ + 1-m？ , the axis of symmetry x=-m, with the image opening facing upward.

Because the maximum value on [- 1, 2] is 4.

Therefore: ① when -m > (- 1+2)/2, that is, m.

② when -m < (- 1+2)/2, that is, m >- 1/2, the maximum value =f(2)=5+4m=4, so: m=- 1/4.

Note: when -m=(- 1+2)/2, that is, m=- 1/2, f(- 1)=f(2)=3, and the maximum value is not 4.