(a) Start with the equation:

\(4 \sin x + \frac{5}{\tan x} + \frac{2}{\sin x} = 0\)

Multiply through by \(\sin x\) to eliminate the fractions:

\(4 \sin^2 x + 5 \cos x + 2 = 0\)

Use the identity \(\sin^2 x = 1 - \cos^2 x\):

\(4(1 - \cos^2 x) + 5 \cos x + 2 = 0\)

Simplify to get:

\(4 - 4 \cos^2 x + 5 \cos x + 2 = 0\)

Rearrange to form a quadratic in \(\cos x\):

\(4 \cos^2 x - 5 \cos x - 6 = 0\)

Thus, \(a = 4, b = -5, c = -6\).

(b) Solve the quadratic equation:

\(4 \cos^2 x - 5 \cos x - 6 = 0\)

Factorize:

\((4 \cos x + 3)(\cos x - 2) = 0\)

Set each factor to zero:

\(4 \cos x + 3 = 0\) or \(\cos x - 2 = 0\)

\(\cos x = -\frac{3}{4}\) or \(\cos x = 2\)

Since \(\cos x = 2\) is not possible, solve \(\cos x = -\frac{3}{4}\):

\(x = \cos^{-1}(-\frac{3}{4})\)

\(x \approx 138.6^\circ\) and \(x \approx 221.4^\circ\)