Question 137593
Your process is right on, but your arithmetic is faulty when you do the tests:


{{{-1.5}}}
{{{(-1.5+1)(-1.5+2)(-1.5+3)=(-.5)(.5)(1.5)<0}}}, (minus times plus times plus is minus), so {{{-1.5}}} produces a false result.


{{{-2.5}}} produces a true result (minus times minus times plus is plus)


{{{-4}}} produces a false result (minus times minus times minus is minus)


{{{0}}} produces a true result -- this one you got right.


So, anything smaller than {{{-3}}} is false,

Anything between {{{-3}}} and {{{-2}}} is true, including {{{-3}}} and {{{-2}}} because of the 'or equal to' part of the original inequality.


Anything between {{{-2}}} and {{{-1}}} is false,


And finally, anything larger than or equal to {{{-1}}} is true.


Hence your interval is <big><b>[{{{-3}}},{{{-2}}}] U [{{{-1}}},{{{infinity}}})</b></big>


The big 'U' stands for 'union' or the set of all elements that are in either or both sets.


A graphical illustration of the situation can be found by multiplying the the three binomials together to get the cubic polynomial {{{x^3+6x^2+11x+6}}} and graph {{{f(x)=x^3+6x^2+11x+6}}}.  The ordered pairs corresponding to those parts of the graph that are either on (equal to) or above (greater than) the x-axis are the solution set of the original inequality.


{{{drawing(600,600,-5,5,-5,5,
grid(1),
graph(600,600,-5,5,-5,5,x^3+6x^2+11x+6)
)}}}