e. $$ \frac{(x+1)\left(x^{2}-36\right)}{x^{2}+8 x+12} $$

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UpStudy AI · Accepted Answer

To simplify the rational expression:

$$
\frac{(x+1)\left(x^{2}-36\right)}{x^{2}+8x+12}
$$

follow these steps:

### 1. **Factor the Numerator and the Denominator**

**Numerator:**

$$
(x + 1)(x^{2} - 36)
$$

Notice that $$ x^{2} - 36 $$ is a **difference of squares**, which can be factored as:

$$
x^{2} - 36 = (x - 6)(x + 6)
$$

So, the factored form of the numerator becomes:

$$
(x + 1)(x - 6)(x + 6)
$$

**Denominator:**

$$
x^{2} + 8x + 12
$$

To factor this quadratic, look for two numbers that multiply to **12** and add up to **8**. These numbers are **6** and **2**.

$$
x^{2} + 8x + 12 = (x + 6)(x + 2)
$$

### 2. **Rewrite the Expression with Factored Forms**

$$
\frac{(x + 1)(x - 6)(x + 6)}{(x + 6)(x + 2)}
$$

### 3. **Simplify by Canceling Common Factors**

Notice that $$ (x + 6) $$ appears in both the numerator and the denominator. You can cancel this common factor:

$$
\frac{(x + 1)(x - 6)\cancel{(x + 6)}}{\cancel{(x + 6)}(x + 2)} = \frac{(x + 1)(x - 6)}{x + 2}
$$

### 4. **Final Simplified Form**

$$
\frac{(x + 1)(x - 6)}{x + 2}
$$

Alternatively, you can expand the numerator if desired:

$$
(x + 1)(x - 6) = x^{2} - 5x - 6
$$

So, the expression can also be written as:

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

**Note:** The simplified form $$\frac{(x + 1)(x - 6)}{x + 2}$$ is generally preferred as it clearly shows the roots and any restrictions on the domain (i.e., $$x \neq -2$$ and $$x \neq -6$$ from the original denominator).
$$ \frac{(x + 1)(x - 6)}{x + 2} $$

e. \( \frac{(x+1)\left(x^{2}-36\right)}{x^{2}+8 x+12} \)

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