(c) \(-\frac{4}{3},\frac{11}{3}\)
\(\frac{5X+6}{(2+X)(1-X)}\) = \(\frac{a}{(2+X)}\) + \(\frac{b}{(1-X)}\)
\(\Rightarrow\) \(\frac{a(1-X)+b(2+X)}{(2+X)(1-X)}\) = \(\frac{5X+6}{(2+X)(1-X)}\)
\(\Rightarrow\)\(\frac{a-aX+2b+bX}{(2+X)(1-X)}\) = \(\frac{5X+6}{(2+X)(1-X)}\)
\(\Rightarrow\) \(\frac{(a+2b)-X(a-b)}{(2+X)(1-X)}\) = \(\frac{5X+6}{(2+X)(1-X)}\)
Equating the coefficient of x and constant term on both the sides of the equation, we get
a + 2b = 6 ......…(i)
a – b = –5 ...........(ii)
Subtracting eqn (ii) from eqn (i), we get
3b = 11
\(\Rightarrow\) b = \(\frac{11}{3}\)
Putting the value of b in (ii), we get
a - \(\frac{11}{3}\) = -5
\(\Rightarrow\)a = -5 + \(\frac{11}{3}\) = \(\frac{-15+11}{3}\) = \(-\frac{4}{3}\).