Correct. Rationalize the denominator by multiplying the numerator and denominator by the denominator's conjugate. Use FOIL.

$\displaystyle\frac{{{\left(\sqrt{{{3}}}+\sqrt{{{7}}}\right)}}}{{{\left(\sqrt{{{3}}}+\sqrt{{{2}}}\right)}}}\cdot\frac{{{\left(\sqrt{{{3}}}-\sqrt{{{2}}}\right)}}}{{{\left(\sqrt{{{3}}}-\sqrt{{{2}}}\right)}}}=\frac{{\sqrt{{{9}}}-\sqrt{{{6}}}+\sqrt{{{21}}}-\sqrt{{{14}}}}}{{\sqrt{{{9}}}-\sqrt{{6}}+\sqrt{{6}}-\sqrt{{{4}}}}}\usepackage{cmbright}\Huge\$

$\displaystyle=\frac{{{3}-\sqrt{{{6}}}+\sqrt{{{21}}}-\sqrt{{{14}}}}}{{{3}-{2}}}={3}-\sqrt{{6}}+\sqrt{{21}}-\sqrt{{14}}\usepackage{cmbright}\Huge\$

The missing number is 14.