To find the LCM (Least Common Multiple) of $x^2 + 6x + 9$ and $x^2 + 8x + 15$, let’s proceed step by step:

1. Factorize each polynomial:

• $x^2 + 6x + 9$
  This is a perfect square trinomial:
  $x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2.$

• $x^2 + 8x + 15$
  Find two numbers that multiply to $15$ and add to $8$: $3$ and $5$.
  $x^2 + 8x + 15 = (x + 3)(x + 5).$

2. Determine the LCM:

The LCM of two expressions includes all unique factors with the highest powers.

• From $x^2 + 6x + 9 = (x + 3)^2$, the factor $x + 3$ appears with power $2$.
• From $x^2 + 8x + 15 = (x + 3)(x + 5)$, the factors are $x + 3$ and $x + 5$.

Combining all unique factors with their highest powers:
$\text{LCM} = (x + 3)^2(x + 5).$

3. Verify the answer choices:

• The correct choice is:
  $(x + 3)(x + 3)(x + 5).$

Final Answer:

The correct option is the first one.

Would you like further explanation or help with similar problems? Here are 5 related questions to explore:

1. What is the general process for finding the LCM of two polynomials?
2. How do you identify perfect square trinomials?
3. How can factoring be used to simplify polynomial expressions?
4. What is the difference between GCF (Greatest Common Factor) and LCM in algebra?
5. How do you multiply and simplify factored polynomial expressions?

Tip: Always start by factoring each polynomial completely when finding the LCM!