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Determine the Lcm of the given polynomials 

To determine the Least Common Multiple (LCM) of the given polynomials, we first need to factor each polynomial and then take the highest power of each factor involved.

### Given Polynomials:
1. \( x^2 + 14x + 49 \)
2. \( x^2 + 10x + 21 \)

### Step 1: Factor each polynomial

1. **Factor \( x^2 + 14x + 49 \):**

   The polynomial is a perfect square trinomial:
   \[
   x^2 + 14x + 49 = (x + 7)^2
   \]

2. **Factor \( x^2 + 10x + 21 \):**

   This polynomial can be factored by finding two numbers that multiply to 21 and add to 10. The factors are 7 and 3:
   \[
   x^2 + 10x + 21 = (x + 7)(x + 3)
   \]

### Step 2: Determine the LCM

The LCM of the polynomials will include each factor at its highest power:

- From \( (x + 7)^2 \)
- From \( (x + 7)(x + 3) \), the factors are \( x + 7 \) and \( x + 3 \).

Therefore, the LCM is:
\[
\text{LCM} = (x + 7)^2(x + 3)
\]

### Final Answer:
\[
\boxed{(x + 7)^2(x + 3)}
\]

Would you like further details on any part of this solution or have any additional questions?

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Here are 5 related questions for further exploration:

1. How do you find the LCM of more than two polynomials?
2. What is the significance of factoring in simplifying polynomial expressions?
3. Can the LCM of two polynomials ever be the same as one of the polynomials?
4. How does the LCM relate to the GCD (Greatest Common Divisor) of polynomials?
5. What are some applications of LCM in solving equations involving fractions?

**Tip:** When factoring trinomials, always check if the quadratic can be expressed as a perfect square trinomial, which makes the process much easier!

Explore how to determine the Least Common Multiple (LCM) of two polynomials x^2 + 14x + 49 and x^2 + 10x + 21. This detailed solution includes factorization techniques and how to identify the highest powers of common factors. Suitable for Grades 10-12.

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