Start by multiplying both sides by the denominator \(3 + 2 \tan x\) to eliminate the fraction:

\(5 + 2 \tan x = (1 + \tan x)(3 + 2 \tan x)\)

Expand the right side:

\(5 + 2 \tan x = 3 + 3 \tan x + 2 \tan x + 2 \tan^2 x\)

Combine like terms:

\(5 + 2 \tan x = 3 + 5 \tan x + 2 \tan^2 x\)

Rearrange to form a quadratic equation:

\(2 \tan^2 x + 3 \tan x - 2 = 0\)

Use the quadratic formula \(\tan x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 2\), \(b = 3\), \(c = -2\):

\(\tan x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2}\)

\(\tan x = \frac{-3 \pm \sqrt{9 + 16}}{4}\)

\(\tan x = \frac{-3 \pm 5}{4}\)

\(\tan x = \frac{2}{4} = 0.5\) or \(\tan x = \frac{-8}{4} = -2\)

For \(\tan x = 0.5\), \(x = \tan^{-1}(0.5) \approx 0.464\) (accept \(0.148\pi\))

For \(\tan x = -2\), \(x = \tan^{-1}(-2) \approx 2.03\) (accept \(0.648\pi\))