To determine the least common denominator (LCD) of the given rational expressions, let's first factor their denominators step-by-step.
### Step 1: Factor the Denominators
1. First Denominator: [tex]\(x^2 + 7x\)[/tex]
To factor this polynomial, we can factor out the greatest common factor (GCF):
[tex]\[
x^2 + 7x = x(x + 7)
\][/tex]
2. Second Denominator: [tex]\(x^2 + 10x + 21\)[/tex]
We need to find two numbers that multiply to 21 and add to 10. The numbers 3 and 7 satisfy this:
[tex]\[
x^2 + 10x + 21 = (x + 3)(x + 7)
\][/tex]
### Step 2: Least Common Denominator (LCD)
To find the LCD, we need to find a common denominator that includes all the factors of both denominators.
- From the first denominator [tex]\(x(x + 7)\)[/tex]:
[tex]\[
\text{Factors: } x, (x + 7)
\][/tex]
- From the second denominator [tex]\((x + 3)(x + 7)\)[/tex]:
[tex]\[
\text{Factors: } (x + 3), (x + 7)
\][/tex]
### Step 3: Combine the Factors
To form the LCD, we take each distinct factor at its highest power:
- The factor [tex]\(x\)[/tex] appears once in the first denominator.
- The factor [tex]\(x + 7\)[/tex] appears once in both denominators.
- The factor [tex]\(x + 3\)[/tex] appears once in the second denominator.
Therefore, the least common denominator will have each of these factors, multiplied together:
[tex]\[
\text{LCD} = x \cdot (x + 7) \cdot (x + 3)
\][/tex]
### Step 4: Verify the Answer
The LCD is:
[tex]\[
x(x + 7)(x + 3)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{C. \, x(x + 7)(x + 3)}
\][/tex]
