If \( x>0 \), then \( \sqrt{x^{3}+x^{2}}-\sqrt{4+4 x}= \) \( \left(x^{2}-4\right) \sqrt{x+1} \) \( \sqrt{\left(x^{2}-4\right)(x+1)} \) \( (x-4) \sqrt{x+1} \) \( (x-2) \sqrt{x+1} \)

To solve \( \sqrt{x^{3}+x^{2}}-\sqrt{4+4x} \) for \( x > 0 \), we start by simplifying each term. 1. **Start with \( \sqrt{x^3 + x^2} \):** We can factor it as \( \sqrt{x^2(x + 1)} = x\sqrt{x + 1} \). 2. **Next, simplify \( \sqrt{4 + 4x} \):** We can factor this as well: \( \sqrt{4(1 + x)} = 2\sqrt{1 + x} \). Putting both simplifications into the expression, we have: \[ x\sqrt{x + 1} - 2\sqrt{1 + x} \] 3. **To further analyze the expression, we can apply the difference of squares:** \[ = \frac{\left((x-2)(\sqrt{x+1})\right)}{\sqrt{x + 1} + 2} \] After simplifying it, we can deduce the valid expression through the relationship and reasoning behind the original formulation. **Final comparison:** From this manipulation, we can see that the left-hand side ultimately describes manipulation involving the roots and terms depending on \( (x - 2)\sqrt{x + 1} \). Based on our analysis: The correct option is \( (x - 2) \sqrt{x + 1} \).