Simplify\(\:\frac{6x^3+9x^2+4x+7}{2x+3}\:and\:3x^2+2+\frac{1}{2x+3}.\:\)
The second expression is simply a rearrangement of the first:
\(\frac{6x^3+9x^2+4x+7}{2x+3} = \frac{3x^2(2x+3)+4x+6+1}{2x+3}= \frac{3x^2(2x+3)+2(2x+3)+1}{2x+3}\\ = \frac{3x^2(2x+3)}{2x+3}+\frac{2(2x+3)}{2x+3}+\frac{1}{2x+3} = 3x^2+2+\frac{1}{2x+3}\)
