To factor the trinomial $x^2 + 4x + 4$, let's follow the steps to identify whether it is a perfect square trinomial and then factor it accordingly.

Step 1: Recognize the trinomial structure

A perfect square trinomial has the form: $a^2 + 2ab + b^2$ Where $a$ and $b$ are numbers or variables.

In this case, the trinomial is $x^2 + 4x + 4$.

Step 2: Identify $a$ and $b$

Compare $x^2 + 4x + 4$ with the perfect square trinomial form $a^2 + 2ab + b^2$:

• $a^2 = x^2$ implies $a = x$.
• The middle term is $2ab = 4x$. Since $a = x$, we can find $b$ by solving: $2b = 4 \quad \Rightarrow \quad b = \frac{4}{2} = 2$

Step 3: Confirm the perfect square structure

The last term $b^2$ in the perfect square trinomial should be $b^2 = 2^2 = 4$. This matches the last term in the original trinomial, confirming that the trinomial is indeed a perfect square trinomial.

Step 4: Write the trinomial as a square of a binomial

Since $a = x$ and $b = 2$, the trinomial can be factored as: $x^2 + 4x + 4 = (x + 2)^2$

Final Answer:

The correct factorization of $x^2 + 4x + 4$ is: $(x + 2)^2$

Would you like more details or have any questions about this process?

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Here are 5 related questions to expand on this topic:

1. How do you factor a trinomial that is not a perfect square?
2. Can you factor $x^2 + 4x + 4$ using a different method, such as completing the square?
3. What is the general formula for factoring the difference of squares?
4. How do you recognize when a trinomial is not factorable over the integers?
5. What are some common mistakes to avoid when factoring trinomials?

Tip: Always check if the trinomial is a perfect square by comparing it with the form $a^2 + 2ab + b^2$. This can save time and ensure accuracy.