College Algebra (10th Edition)

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