It appears that you combined the original numerators while changing the denominators to the correct LCD. Instead, rewrite each term as an equivalent fraction before adding or subtracting:

$\displaystyle\frac{{{7}{{x}}^{{2}}}}{{{{x}}^{{2}}-{49}}}-\frac{{{7}{x}}}{{{x}+{7}}}+\frac{{7}}{{{x}-{7}}}=\frac{{{7}{{x}}^{{2}}}}{{{\left({x}+{7}\right)}{\left({x}-{7}\right)}}}-\frac{{{\left({7}{x}\right)}{\left({x}-{7}\right)}}}{{{\left({x}+{7}\right)}{\left({x}-{7}\right)}}}+\frac{{{7}{\left({x}+{7}\right)}}}{{{\left({x}-{7}\right)}{\left({x}+{7}\right)}}}\usepackage{cmbright}\Huge\$

Then simplify and combine the numerators.