start rewriting the square term and the linear term
[tex]\frac{2x^3+2x^2+7x^2+7x+3x+3}{2x+1}[/tex]take the common factor
[tex]\frac{2x^2(x+1)+7x\cdot(x+1)+3\cdot(x+1)}{2x+1}[/tex]factor the expression
[tex]\frac{(x+1)\cdot(2x^2+7x+3)}{2x+1}[/tex]rewrite the expression
[tex]\frac{(x+1)\cdot(2x^2+6x+x+3)}{2x+1}[/tex]take common factor
[tex]\frac{(x+1)\cdot(2x(x+3)+x+3)}{2x+1}[/tex]factor the expression
[tex]\frac{(x+1)\cdot(x+3)(2x+1)}{2x+1}[/tex]simplify
[tex]\begin{gathered} (x+1)(x+3) \\ x^2+x+3x+3 \\ x^2+4x+3 \end{gathered}[/tex]