Square both sides: $\displaystyle{{\left(\sqrt{{{x}+{5}}}\right)}}^{{2}}={{\left({x}-{7}\right)}}^{{2}}\usepackage{cmbright}\Huge\$

Simplify and multiply: $\displaystyle{x}+{5}={{x}}^{{2}}-{14}{x}+{49}\usepackage{cmbright}\Huge\$

Subtract $\displaystyle{x}+{5}\usepackage{cmbright}\Huge\$ from both sides: $\displaystyle{0}={{x}}^{{2}}-{15}{x}+{44}\usepackage{cmbright}\Huge\$

Factor and solve: $\displaystyle{0}={\left({x}-{4}\right)}{\left({x}-{11}\right)}\usepackage{cmbright}\Huge\$ , so $\displaystyle{x}={4}\usepackage{cmbright}\Huge\$ and $\displaystyle{x}={11}\usepackage{cmbright}\Huge\$

Check $\displaystyle{x}={4}\usepackage{cmbright}\Huge\$ : $\displaystyle\sqrt{{{4}+{5}}}={4}-{7}\usepackage{cmbright}\Huge\$ ? $\displaystyle\sqrt{{9}}\ne-{3}\usepackage{cmbright}\Huge\$ extraneous solution
Check $\displaystyle{x}={11}\usepackage{cmbright}\Huge\$ : $\displaystyle\sqrt{{{11}+{5}}}={11}-{7}\usepackage{cmbright}\Huge\$ ? $\displaystyle\sqrt{{16}}={4}\usepackage{cmbright}\Huge\$ TRUE

Check $\displaystyle{x}={4}\usepackage{cmbright}\Huge\$ :	$\displaystyle\sqrt{{{4}+{5}}}={4}-{7}\usepackage{cmbright}\Huge\$ ?	$\displaystyle\sqrt{{9}}\ne-{3}\usepackage{cmbright}\Huge\$	extraneous solution
Check $\displaystyle{x}={11}\usepackage{cmbright}\Huge\$ :	$\displaystyle\sqrt{{{11}+{5}}}={11}-{7}\usepackage{cmbright}\Huge\$ ?	$\displaystyle\sqrt{{16}}={4}\usepackage{cmbright}\Huge\$	TRUE