To factorize the given expression \(100x^2 + 25 + 1\), let's go through it step-by-step.

1. The expression is \(100x^2 + 25 + 1\).

2. Combine like terms: \(100x^2 + 26\).

Now, let's see if this can be factored.

We can use the method of completing the square or check if it is a perfect square trinomial, but in this case, the expression is just \(100x^2 + 26\). 

To factorize:

1. Factor out the greatest common factor (GCF) if there is one. Here, the GCF of \(100x^2\) and \(26\) is 2.

   \[
   100x^2 + 26 = 2(50x^2 + 13)
   \]

2. The expression inside the parentheses, \(50x^2 + 13\), does not factor further over the integers.

So, the factored form of \(100x^2 + 26\) is:

\[
2(50x^2 + 13)
\]

If you have any further questions or need more details, please let me know!

Here are some related questions to expand your understanding:

1. How do you determine the greatest common factor (GCF) of a polynomial?
2. What is a perfect square trinomial, and how do you identify one?
3. Can the expression \(50x^2 + 13\) be factored over the complex numbers?
4. What methods can be used to factor polynomials?
5. How do you factor polynomials using the method of completing the square?
6. What is the difference between factorization over integers and factorization over complex numbers?
7. How do you verify if a polynomial is prime and cannot be factored further?
8. What are the common techniques for factoring quadratic expressions?

**Tip:** Always check if there is a greatest common factor (GCF) in all terms of a polynomial before attempting other factorization methods. This simplifies the process and reduces errors.

Explore how to factorize the expression 100x^2 + 25 + 1 using detailed step-by-step instructions. This problem covers polynomial factoring techniques and is suitable for high school students in Grades 10-12.

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