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

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UpStudy AI · Accepted Answer

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!
The two rational expressions are: 1. $$ \frac{x + 3}{(x + 9)(x + 2)} $$ 2. $$ \frac{-2x}{(x + 9)(x - 3)} $$ Both expressions are already in their simplest forms.

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

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