Answer
$\displaystyle \frac{4x^{2}+1}{3x^{2}-1}$
Work Step by Step
$\displaystyle \frac{4+\frac{1}{x^{2}}}{3-\frac{1}{x^{2}}}=(4+\frac{1}{x^{2}})\div(3-\frac{1}{x^{2}})$
$=(\displaystyle \frac{4x^{2}}{x^{2}}+\frac{1}{x^{2}})\div(\frac{3x^{2}}{x^{2}}-\frac{1}{x^{2}})$
$=\displaystyle \frac{4x^{2}+1}{x^{2}}\div\frac{3x^{2}-1}{x^{2}}$
... division = multiplication with the reciprocal,
$=\displaystyle \frac{4x^{2}+1}{x^{2}}\times\frac{x^{2}}{3x^{2}-1}$
... $x^{2}$ cancels,
$=\displaystyle \frac{4x^{2}+1}{3x^{2}-1}$
