Correct answer: 3
Case I: \(\mathrm{x} \geq-5\)
\(\mathrm{x}^{2}+5 \mathrm{x}+2 \mathrm{x}+12=0\)
\(\mathrm{x}^{2}+7 \mathrm{x}+12=0\)
\(\mathrm{x}=-3,-4\)
Case II: \(-7<\mathrm{x}<-5\)
\(-x^{2}-5 x+2 x+14-2=0\)
\(-x^2 -3x +12 = 0\)
\(\mathrm{x}=\frac{-3 \pm \sqrt{9+48}}{2}\)
\(=\frac{-3 \pm \sqrt{57}}{2}\)
\(x=\frac{-3-\sqrt{57}}{2}, \frac{-3+\sqrt{57}}{2}\) (rejected)
Case III: \(x \leq-7\)
\(-\mathrm{x}^{2}-5 \mathrm{x}-2 \mathrm{x}-14-2=0\)
\(\mathrm{x}^{2}+7 \mathrm{x}+16=0\)
\(\mathrm{D}=49-64<0\)
No solutions
No. of solutions = 3
