Correct answer: 3
Case-1
\(x \geq 0\)
\(x^2+2 x-5 x-5-1=0\)
\(x^2-3 x-6=0\)
\( x=\frac{3 \pm \sqrt{9+24}}{2}=\frac{3 \pm \sqrt{33}}{2}\)
One positive root.
Case-2
\(-1 \leq x<0\)
\(-x^2-2 x-5 x-5-1=0\)
\( x^2+7 x+6=0\)
\((x+6)(x+1)=0\)
\(x=-1\)
One root in range.
Case-3
\(-2 \leq x<-1\)
\(x^2-2 x+5 x+5-1=0\)
\(x^2-3 x-4=0\)
\((x-4)(x+1)=0\)
No root in range.
Case-4
\(x < -2\)
\(x^2 + 7x + 4= 0\)
\(x = \frac{-7 \pm \sqrt{49 - 16}}2 = \frac{7 \pm \sqrt {33}}2\)
One root in range.
Total number of distinct roots are 3.
