Correct Answer: Option B

Explanation

\(\frac{x^{2} + x + 4}{(1 - x)(x^{2} + 1)} = \frac{A}{1 -  x} + \frac{Bx + C}{x^{2} + 1}\)

= \(\frac{A(x^{2} + 1) + (Bx + C)(1 - x)}{(1 - x)(x^{2} + 1)}\)

\(\implies x^{2} + x + 4 = A(x^{2} + 1) + (Bx + C)(1 - x)\)

\(x^{2} + x + 4 = Ax^{2} + A + Bx - Bx^{2} - Cx + C\)

\(\implies (A - B)x^{2} = x^{2}; A - B = 1 ...... (i)\)

\((B - C)x = x; B - C = 1 ..... (ii)\)

\(A + C = 4 ...... (iii)\)

Solving the above simultaneous equations by any of the known methods, we get

\(A = 3, B = 2, C = 1\)

\(\therefore  \frac{x^{2} + x + 4}{(1 - x)(x^{2} + 1)} = \frac{3}{1 - x} + \frac{2x + 1}{x^{2} + 1}\)

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