https://file.aiquickdraw.com/mathbot/file/17273018757055ru10p7h.jpg

Let's break down the factorization step by step.

The expression given is:

\[
x^2 + 5x - 1 = (2k)^2 + 5(2k) - 1
\]
This simplifies to:

\[
= 4k^2 + 10k - 1
\]

Now, the next step in the factorization shows:

\[
= 2(2k^2 + 5k - 1) + 1
\]

To understand how the factorization was performed:

1. The expression \( 4k^2 + 10k - 1 \) was first factored by grouping terms in a way that separates a common factor of 2. Notice that the terms \( 4k^2 + 10k \) can be divided by 2, which gives \( 2(2k^2 + 5k) \).

2. The remaining part of the expression, \(-1\), is kept outside the parentheses initially.

3. To simplify further, this results in:
   \[
   2(2k^2 + 5k - 1) + 1
   \]

This process simplifies the quadratic expression into a form that prepares it for further analysis or solving.

### Would you like more detailed clarification or explanation on any part of this?

Here are 5 related questions for practice:

1. How do you factor a quadratic expression in general?
2. Can you explain why the factor of 2 was taken out in this case?
3. What methods can be used to solve quadratic equations after factoring?
4. How do you identify common factors in algebraic expressions?
5. What are other methods for simplifying a quadratic expression?

**Tip:** When factoring, always check for common factors among the terms to simplify the process.

How was the factorization performed for the quadratic expression x^2 + 5x - 1?

This page explains how to factor and simplify the quadratic expression x^2 + 5x - 1, showing step-by-step transformations from one form to another. Key algebraic concepts such as quadratic expressions and factorization are covered in detail. Suitable for students in Grades 8-10.

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