Given:
$$x|x+3|+|x-1|-2=0$$
Case 1: $$ x < -3$$
If $$x < -3$$ then $$|x+3| = -(x + 3)$$ and $$|x -1| = -(x -1)$$
$$\Rightarrow$$ $$-x(x + 3) -(x-1) - 2 = 0$$
$$\Rightarrow$$ $$ -x^2 -3x - x + 1 - 2 = 0$$
$$\Rightarrow$$ $$ x^2 + 4x + 1 = 0$$
$$\Rightarrow$$ $$x^2 + 4x + 4 = -1 + 4$$
$$\Rightarrow$$ $$(x + 2)^2 = 3$$
$$\Rightarrow$$ $$ (x + 2) = \sqrt{3} $$ OR $$(x +2) = -\sqrt{3}$$
$$\Rightarrow$$ $$ x = - 2 + \sqrt{3} \approx -0.268$$ OR $$x = -2 -\sqrt{3} \approx -3.732$$
But $$x < -3$$ $$\Rightarrow$$ $$x = -2 -\sqrt{3}$$ $$\Rightarrow$$ One solution from this case
Case 2: $$ -3 \leq x < 1$$
If $$ -3 \geq x < 1$$ then $$|x+3| = (x + 3)$$ and $$|x -1| = -(x -1)$$
$$\Rightarrow$$ $$x(x+3) -(x -1) - 2 = 0$$
$$\Rightarrow$$ $$x^2 + 2x - 1 = 0$$
$$\Rightarrow$$ $$(x + 1)^2 = 2$$
$$\Rightarrow$$ $$(x +1) = \sqrt{2}$$ OR $$(x + 1) = -\sqrt{2}$$
$$\Rightarrow$$ $$ x = -1 + \sqrt{2} \approx 0.414$$ OR $$x = -1 - \sqrt{2} \approx -2.414$$
Both of these satisfy the given condition
$$\Rightarrow$$ Two solutions from this case
Case 3: $$ x \geq 1$$
If $$x \geq 1$$ then $$|x+3| = (x + 3)$$ and $$|x -1| = (x -1)$$
$$\Rightarrow$$ $$ x(x +3) + (x -1) - 2 = 0$$
$$\Rightarrow$$ $$x^2 + 4x = 3$$
$$\Rightarrow$$ $$x^2 + 4x + 4 = 7$$
$$\Rightarrow$$ $$ (x + 2)^2 = 7$$
$$\Rightarrow$$ $$(x + 2) = \sqrt{7}$$ OR $$(x + 2) = -\sqrt{7}$$
$$\Rightarrow$$ $$x = -2 + \sqrt{7} \approx 0.646$$ OR $$x = -2 -\sqrt{7} \approx -4.646$$
None of these satisfies the condition. Thus zero solutions from this case.
Hence there are total of 3 solutions
$$x = -2 -\sqrt{3}, x = -1 + \sqrt{2}, x = -1 - \sqrt{2}$$
Hence, option C is the correct choice.
91- 6303239042
