Given two polynomials f(x) and g(x), the division between them can be represented by:
[tex]\frac{f(x)}{g(x)}=q(x)+\frac{r(x)}{g(x)}[/tex]Where q(x) is the quotient and r(x) is the remainder. From the problem, we identify:
[tex]\begin{gathered} f(x)=3x^3+11x^2-12x-19 \\ \\ g(x)=3x+2 \end{gathered}[/tex]Then, using the long division method:
Finally:
[tex]\begin{gathered} r(x)=-7 \\ q(x)=x^2+3x-6 \end{gathered}[/tex]And the result is:
[tex]\frac{f(x)}{g(x)}=x^2+3x-6-\frac{7}{3x+2}[/tex]