1. \( \left(\frac{1}{x}+2\right)\left(\frac{3}{x}-1\right) \) 2. \( (a b+c)\left(a^{2} b^{2}+c^{2}\right)(a b-c) \) 3. \( (a-2)\left(a^{3}+8\right)\left(a^{2}+2 a+4\right. \) 4. \( \left(a^{x}-b^{y}\right)\left(a^{x}+b^{y}\right) \)

Sure, let's simplify and analyze each of the provided expressions step by step. --- ### 1. \( \left(\frac{1}{x} + 2\right)\left(\frac{3}{x} - 1\right) \) **Simplification:** Multiply the two binomials using the distributive property (FOIL method): \[ \begin{align*} \left(\frac{1}{x} + 2\right)\left(\frac{3}{x} - 1\right) &= \frac{1}{x} \cdot \frac{3}{x} + \frac{1}{x} \cdot (-1) + 2 \cdot \frac{3}{x} + 2 \cdot (-1) \\ &= \frac{3}{x^2} - \frac{1}{x} + \frac{6}{x} - 2 \\ &= \frac{3}{x^2} + \left(-\frac{1}{x} + \frac{6}{x}\right) - 2 \\ &= \frac{3}{x^2} + \frac{5}{x} - 2 \end{align*} \] **Final Simplified Form:** \[ \frac{3}{x^2} + \frac{5}{x} - 2 \] --- ### 2. \( (a b + c)\left(a^{2} b^{2} + c^{2}\right)(a b - c) \) **Simplification:** Notice that \( (ab + c)(ab - c) \) is a difference of squares: \[ (ab + c)(ab - c) = (ab)^2 - c^2 = a^2b^2 - c^2 \] Now, multiply this result by the remaining factor \( a^{2} b^{2} + c^{2} \): \[ (a^2b^2 - c^2)(a^2b^2 + c^2) = (a^2b^2)^2 - (c^2)^2 = a^4b^4 - c^4 \] **Final Simplified Form:** \[ a^4b^4 - c^4 \] --- ### 3. \( (a - 2)\left(a^{3} + 8\right)\left(a^{2} + 2 a + 4\right) \) **Simplification:** Firstly, recognize that \( a^3 + 8 \) is a sum of cubes: \[ a^3 + 8 = a^3 + 2^3 = (a + 2)(a^2 - 2a + 4) \] Now, substitute this back into the original expression: \[ (a - 2)(a + 2)(a^2 - 2a + 4)\left(a^{2} + 2 a + 4\right) \] Notice that \( a^2 - 2a + 4 \) and \( a^2 + 2a + 4 \) are quadratic terms that don’t factor further over the real numbers. **Final Factored Form:** \[ (a - 2)(a + 2)(a^2 - 2a + 4)(a^2 + 2a + 4) \] *Alternatively*, if you prefer to leave the expression partially factored: \[ (a - 2)(a^3 + 8)(a^2 + 2a + 4) \] --- ### 4. \( \left(a^{x} - b^{y}\right)\left(a^{x} + b^{y}\right) \) **Simplification:** This follows the pattern of the difference of squares: \[ (a^x - b^y)(a^x + b^y) = (a^x)^2 - (b^y)^2 = a^{2x} - b^{2y} \] **Final Simplified Form:** \[ a^{2x} - b^{2y} \] --- If you have any specific questions about these simplifications or need further assistance, feel free to ask!