Divide and simplify the following rational expression. $$ \frac{x^{2}+x-6}{x^{2}+5 x+6} \div \frac{x-6}{x+2} $$

Question

UpStudy AI · Accepted Answer

To simplify the given rational expression:

$$
\frac{x^{2}+x-6}{x^{2}+5x+6} \div \frac{x-6}{x+2}
$$

**Step 1: Rewrite the Division as Multiplication**

Dividing by a fraction is the same as multiplying by its reciprocal:

$$
\frac{x^{2}+x-6}{x^{2}+5x+6} \times \frac{x+2}{x-6}
$$

**Step 2: Factor the Polynomials**

Factor both the numerator and the denominator of each fraction.

- **Factor $$ x^{2} + x - 6 $$:**
  
  We look for two numbers that multiply to $$-6$$ and add to $$1$$. These numbers are $$3$$ and $$-2$$.

$$
  x^{2} + x - 6 = (x + 3)(x - 2)
  $$

- **Factor $$ x^{2} + 5x + 6 $$:**
  
  We look for two numbers that multiply to $$6$$ and add to $$5$$. These numbers are $$2$$ and $$3$$.

$$
  x^{2} + 5x + 6 = (x + 2)(x + 3)
  $$

**Step 3: Substitute the Factored Forms**

$$
\frac{(x + 3)(x - 2)}{(x + 2)(x + 3)} \times \frac{x + 2}{x - 6}
$$

**Step 4: Simplify by Canceling Common Factors**

- **Cancel $$ (x + 3) $$ from the numerator and denominator:**

$$
\frac{(x - 2)}{(x + 2)} \times \frac{x + 2}{x - 6}
$$

- **Cancel $$ (x + 2) $$ from the numerator and denominator:**

$$
\frac{x - 2}{x - 6}
$$

**Final Simplified Expression:**

$$
\boxed{\dfrac{x-2}{x-6}}
$$

**Note:** The original expression is undefined for $$ x = -3, -2, 6 $$ because these values would make the denominators zero.

**Answer:** 
\boxed{\dfrac{x-2}{x-6}}
The simplified expression is $$ \frac{x - 2}{x - 6} $$.

Divide and simplify the following rational expression.