$\begin{array}{l} 7x + 5\sqrt {x + 1} = \sqrt {12 - x} + 28\\ \Leftrightarrow 7x + 5\sqrt {x + 1} - \sqrt {12 - x} - 28 = 0\\ \Leftrightarrow 7x - 21 + 5\left( {\sqrt {x + 1} - 2} \right) + 3 - \sqrt {12 - x} = 0\\ \Leftrightarrow 7\left( {x - 3} \right) + 5.\dfrac{{x - 3}}{{\sqrt {x + 1} + 2}} + \dfrac{{9 - \left( {12 - x} \right)}}{{3 + \sqrt {12 - x} }} = 0\\ \Leftrightarrow 7\left( {x - 3} \right) + 5\dfrac{{\left( {x - 3} \right)}}{{\sqrt {x + 1} + 2}} + \dfrac{{x - 3}}{{3 + \sqrt {12 - x} }} = 0\\ \Leftrightarrow \left( {x - 3} \right)\left( {\underbrace {7 + \dfrac{5}{{\sqrt {x + 1} + 2}} + \dfrac{1}{{3 + \sqrt {12 - x} }}}_{ > 0}} \right) = 0\\ \Leftrightarrow x = 3\\ \Rightarrow S = \left\{ 3 \right\} \end{array}$
