The equation of curves through the point \( (1,0) \) and whose slope is \( \frac{y-1}{x^{2}+x} \) is
\(\frac{dy}{dx} = \frac{y-1}{x(x+1)} \Rightarrow \frac{dy}{y-1} = \frac{dx}{x(x+1)} = \left(\frac{1}{x} - \frac{1}{x+1}\right)dx\)
Integrate: \(\ln|y-1| = \ln\left|\frac{x}{x+1}\right| + C \Rightarrow y - 1 = k\frac{x}{x+1}\)
It passes through (1, 0): \(-1 = \frac{k}{2} \Rightarrow k = -2\).
\((y-1)(x+1) = -2x \Rightarrow (y-1)(x+1) + 2x = 0\)
Answer: (A)
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