\(a_{n} = 1.\)

The divider of \(a_{0} is 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96, 192.\)

Rational Zero \(= \pm \frac{1,2,3,4,6,8,12,16,24,32,48,96,192}{1}\)

The root of the equation is - 3 by trial and error method then, the factor is \(x + 3\). Now, divide the provided equation with calculated factor.

\(x^{4} - x^{3} + 4x^{2} - 16x - 192 = \frac{x^{4}-x^{3}+4x^{2}-16x-192}{x+3}\)

\(= (x + 3)(x^{3} - 4x2 + 16x - 64)\)

Now, further factorize the above equation.

\((x + 3)(x^{3} - 4x^{2} + 16x - 64) = 0\)

\((x + 3)((x^{3} - 4x^{2}) + (16x - 64)) = 0\)

\((x + 3)(x^{2}(x - 4) + 16(x - 4)) = 0\)

\((x + 3)(x - 4)(x^{2} + 16) = 0\)

\(x = -3, 4, \pm 4i\)

Answer: Therefore, the real solution of the provided equation is \(x = -3, 4.\)