$\begin{array}{l} \sqrt {2{x^2} + 5x + 12} + \sqrt {2{x^2} + 3x + 2} = x + 5\\ \Leftrightarrow a + b = \dfrac{{{a^2} - {b^2}}}{2}\left( {Do\,x + 5 = \dfrac{{{a^2} - {b^2}}}{2},\sqrt {2{x^2} + 5x + 12} = a,\sqrt {2{x^2} + 3x + 2} = b} \right)\\ \Leftrightarrow \left( {a - b} \right)\left( {a + b} \right) = 2\left( {a + b} \right)\\ \Leftrightarrow \left( {a + b} \right)\left( {a - b - 2} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l} a + b = 0(L)\\ a = b + 2 \end{array} \right.\\ a = b + 2 \Leftrightarrow a + b = x + 5 \Leftrightarrow 2b + 2 = x + 5\\ \Leftrightarrow 2\sqrt {2{x^2} + 3x + 2} = x + 3\\ \Leftrightarrow 8{x^2} + 12x + 8 = {x^2} + 6x + 9\\ \Leftrightarrow \left( {7x - 1} \right)\left( {x + 1} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l} x = \dfrac{1}{7}\\ x = - 1 \end{array} \right. \end{array}$
