we have the expression
[tex]\frac{12x^2+31x+9}{3x^2-5x-2}[/tex]Remember that
the denominator can not be equal to zero
so
Find the roots of denominator
using a graphing tool
[tex]3x^2-5x-2=0[/tex]the roots are
x=-1/3 and x=2
that means
[tex]3x^2-5x-2=3(x+\frac{1}{3})(x-2)[/tex]
