\( \frac{x+3}{x^{2}+11 x+18} \) and \( \frac{-2 x}{x^{2}+6 x-27} \)

Certainly! Let's analyze and simplify the two rational expressions you've provided: 1. **First Expression:** \[ \frac{x + 3}{x^{2} + 11x + 18} \] **Factor the Denominator:** \[ x^{2} + 11x + 18 = (x + 9)(x + 2) \] **Simplified Form:** \[ \frac{x + 3}{(x + 9)(x + 2)} \] **Check for Common Factors:** - The numerator \( x + 3 \) does not factor further and does not share any common factors with the denominator. Therefore, the expression is already in its simplest form. 2. **Second Expression:** \[ \frac{-2x}{x^{2} + 6x - 27} \] **Factor the Denominator:** \[ x^{2} + 6x - 27 = (x + 9)(x - 3) \] **Simplified Form:** \[ \frac{-2x}{(x + 9)(x - 3)} \] **Check for Common Factors:** - The numerator \( -2x \) does not share any common factors with the denominator other than constants. The expression is already simplified. --- ### **Additional Operations (Optional)** If you're interested in performing operations with these two expressions, here's how you can add them: **Adding the Two Expressions:** \[ \frac{x + 3}{(x + 9)(x + 2)} + \frac{-2x}{(x + 9)(x - 3)} \] **Find a Common Denominator:** The common denominator is \( (x + 9)(x + 2)(x - 3) \). **Express Each Fraction with the Common Denominator:** \[ \frac{(x + 3)(x - 3)}{(x + 9)(x + 2)(x - 3)} + \frac{-2x(x + 2)}{(x + 9)(x + 2)(x - 3)} \] **Combine the Numerators:** \[ \frac{(x + 3)(x - 3) - 2x(x + 2)}{(x + 9)(x + 2)(x - 3)} \] **Simplify the Numerator:** \[ (x + 3)(x - 3) = x^2 - 9 \] \[ -2x(x + 2) = -2x^2 - 4x \] \[ \text{Combined Numerator} = x^2 - 9 - 2x^2 - 4x = -x^2 - 4x - 9 \] **Final Combined Expression:** \[ \frac{-x^2 - 4x - 9}{(x + 9)(x + 2)(x - 3)} \] This combined expression represents the sum of the two original rational expressions. --- ### **Domain Considerations** For both expressions, it's important to note the values of \( x \) that make the denominator zero, as these are excluded from the domain. 1. **First Expression:** \[ (x + 9)(x + 2) = 0 \implies x = -9, -2 \] - **Domain:** All real numbers except \( x = -9 \) and \( x = -2 \). 2. **Second Expression:** \[ (x + 9)(x - 3) = 0 \implies x = -9, 3 \] - **Domain:** All real numbers except \( x = -9 \) and \( x = 3 \). 3. **Combined Expression:** \[ (x + 9)(x + 2)(x - 3) = 0 \implies x = -9, -2, 3 \] - **Domain:** All real numbers except \( x = -9, -2, 3 \). --- If you have a specific operation or question in mind regarding these expressions, feel free to provide more details!