1. **State the problem:** Find the square root of the expression $$x^2 + \frac{1}{x^2} + 2\left(x + \frac{1}{x}\right) + 3$$ by factorization. 2. **Rewrite the expression:** Group terms to see if it can be expressed as a perfect square. $$x^2 + \frac{1}{x^2} + 2\left(x + \frac{1}{x}\right) + 3$$ 3. **Recall the identity:** $$\left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}$$ 4. **Express parts of the expression using the identity:** $$x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2$$ 5. **Substitute back:** $$\left(x + \frac{1}{x}\right)^2 - 2 + 2\left(x + \frac{1}{x}\right) + 3$$ 6. **Simplify constants:** $$\left(x + \frac{1}{x}\right)^2 + 2\left(x + \frac{1}{x}\right) + 1$$ 7. **Recognize the perfect square trinomial:** $$\left(x + \frac{1}{x} + 1\right)^2$$ 8. **Therefore, the square root is:** $$\sqrt{\left(x + \frac{1}{x} + 1\right)^2} = \left|x + \frac{1}{x} + 1\right|$$ **Final answer:** $$\boxed{\left|x + \frac{1}{x} + 1\right|}$$