Express \(2 \ln x + \ln 3\) as a single logarithm.
Hence, given that \(x\) satisfies the equation
$$2 \ln x + \ln 3 = \ln ( 5 x + 2 )$$
show that \(x\) is a root of the quadratic equation \(3 x ^ { 2 } - 5 x - 2 = 0\).
Solve this quadratic equation, explaining why only one root is a valid solution of
$$2 \ln x + \ln 3 = \ln ( 5 x + 2 ) .$$