When m=1, x^3+2x^2+3x=x^2-2x-3; x^3+x^2+5x+3=0.
When x=0 the cubic has a value of 3. Because all the signs are plus, when x>0 the cubic stays positive.
When x=-1 the cubic has a value of -2, so there is a root between 0 and -1.
When x<-1 it has negative values because x^3 is always more negative than x^2 and the combined negative values exceed -3 so the whole cubic remains negative.
When m=-2/3, -2x^3/3-4x^2/3-2x=x^2-2x-3; 2x^3/3+7x^2/3-3=0; 2x^3+7x^2-9=0.
This cubic has a root at x=1 because 2+7-9=0.
We can use synthetic division to find the quadratic:
1 | 2 7 0 -9
.....2 2 9 9
.....2 9 9 | 0 = 2x^2+9x+9=0=(2x+3)(x+3). This shows there are three roots: x=1, -3/2, -3.