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The given expression is:

\[
x^2 + 22x + 121
\]

This is a quadratic expression. To factor it, we can check if it fits the form of a perfect square trinomial, which has the structure:

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

In this case:

- \( a^2 = x^2 \), so \( a = x \)
- \( b^2 = 121 \), so \( b = 11 \)
- Check if the middle term \( 2ab = 2(x)(11) = 22x \), which is correct.

Thus, the expression is a perfect square trinomial, and it factors as:

\[
(x + 11)^2
\]

### Final Factored Form:
\[
x^2 + 22x + 121 = (x + 11)^2
\]

Would you like further explanation or have any questions?

### Related Questions:
1. How can you identify a perfect square trinomial?
2. What are some other examples of perfect square trinomials?
3. How do you factor a non-perfect square trinomial?
4. What are the general steps for factoring quadratic expressions?
5. Can a quadratic expression always be factored?

**Tip:** When factoring trinomials, always check if it's a perfect square trinomial by comparing it to the form \( a^2 + 2ab + b^2 \).

Learn how to factor the quadratic expression x^2 + 22x + 121 using the perfect square trinomial method. This step-by-step solution explains algebraic concepts for students in Grades 8-10.

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