If a + b = 10 and $$\sqrt {\frac{a}{b}} - 13 = - \sqrt {\frac{b}{a}} - 11$$     then what is the value of 3ab + 4a² + 5b²?

$$\eqalign{ & {\text{Let }}\sqrt {\frac{a}{b}} = x \cr & \therefore \,x - 13 = \frac{{ - 1}}{x} - 11 \cr & x + \frac{1}{x} = 2 \cr & \therefore \,x = 1 \cr & \sqrt {\frac{a}{b}} = 1{\text{ and }}a + b = 10 \cr & \therefore \,a = b = 5 \cr & 3ab + 4{a^2} + 5{b^2} \cr & = 3{a^2} + 4{a^2} + 5{a^2} \cr & = 12{a^2} \cr & = 12 \times 25 \cr & = 300 \cr} $$