Question 114407
If you expanded the right side of the equation you would find that the highest order term would be {{{2x^3}}}.  This means that you have a polynomial equation of degree 3, and this means that there are exactly three roots or zeros for this equation.


Expanding the equation a little you get:


{{{y=(2x+3)(x-1)(x-1)}}}


You should be able to see that y will be zero if and only if:


{{{2x+3=0}}} or
{{{x-1=0}}} or 
{{{x-1=0}}}


Hence, the three roots are {{{x=-3/2}}} or {{{x=1}}} or {{{x=1}}}.  Note that just because two of the roots are equal doesn't change the fact that there are three of them.


Graphically, this is going to look like a sloppy letter 'N'.  The graph will cross the x-axis at {{{-3/2}}} and the x-axis will be tangent to a local minimum point at (1, 0).  It will also have a local maximum at ({{{-2/3}}},{{{5/27}}}).  The graph will intercept the y-axis at 3.


{{{graph(600,600,-5,5,-5,5,(2x+3)(x-1)^2)}}}