The problem shown in the image asks to identify the least common denominator (LCD) of two rational expressions:

$\frac{x}{x^2 + 12x + 32} \quad \text{and} \quad \frac{4x}{x^2 + 17x + 72}$

To solve this, we need to factor both quadratic expressions in the denominators.

Step 1: Factor the first denominator

The first denominator is $x^2 + 12x + 32$.

We need to find two numbers that multiply to $32$ and add to $12$. These numbers are $4$ and $8$, so:

$x^2 + 12x + 32 = (x + 4)(x + 8)$

Step 2: Factor the second denominator

The second denominator is $x^2 + 17x + 72$.

We need to find two numbers that multiply to $72$ and add to $17$. These numbers are $8$ and $9$, so:

$x^2 + 17x + 72 = (x + 8)(x + 9)$

Step 3: Identify the Least Common Denominator (LCD)

The least common denominator is the product of all distinct factors, taking the highest power of each:

• From the first denominator: $(x + 4)(x + 8)$
• From the second denominator: $(x + 8)(x + 9)$

The LCD must include each distinct factor:

$\text{LCD} = (x + 4)(x + 8)(x + 9)$

So, the least common denominator is $(x + 4)(x + 8)(x + 9)$.

Would you like more details on this solution or any other questions?

Here are 5 related questions that might help you explore this topic further:

1. How do you factor a quadratic expression in general?
2. What are the steps to find the least common denominator for three or more rational expressions?
3. How would the solution change if one of the denominators was a higher-degree polynomial?
4. How can we simplify a complex rational expression once we have the LCD?
5. What is the process of performing operations (addition, subtraction) on rational expressions with different denominators?

Tip: Always double-check your factors to ensure they multiply back to the original expression. This helps avoid errors in complex algebraic operations.