I suppose you mean
[tex]\dfrac{3x^3+4x^2+6x+9}{x^2+3x+7}[/tex]
[tex]3x^3=3x\cdot x^2[/tex], and if we multiply [tex]x^2+3x+7[/tex] by [tex]3x[/tex] we get [tex]3x^3+9x^2+21x[/tex]. Subtracting this from the numerator gives a remainder of [tex]-5x^2-15x+9[/tex].
[tex]-5x^2=-5\cdot x^2[/tex], and multiplying [tex]x^2+3x+7[/tex] by [tex]-5[/tex] gives [tex]-5x^2-15x-35[/tex]. Subtracting this from the previous remainder gives a new remainder of [tex]44[/tex].
[tex]44[/tex] has no remaining factors of [tex]x^2[/tex] in it, so we're done, and
[tex]\dfrac{3x^3+4x^2+6x+9}{x^2+3x+7}=\underbrace{3x-5}_{\rm quotient}+\dfrac{\overbrace{44}^{\rm remainder}}{x^2+3x+7}[/tex]
