To find the product of \((2n - 1)(5n + 1)\), we can use the distributive property, often referred to as FOIL (First, Outer, Inner, Last) in the case of binomials.

\[
(2n - 1)(5n + 1)
\]

1. **First**: Multiply the first terms of each binomial.
   \[
   2n \times 5n = 10n^2
   \]

2. **Outer**: Multiply the outer terms.
   \[
   2n \times 1 = 2n
   \]

3. **Inner**: Multiply the inner terms.
   \[
   -1 \times 5n = -5n
   \]

4. **Last**: Multiply the last terms of each binomial.
   \[
   -1 \times 1 = -1
   \]

Now, combine all the terms:

\[
10n^2 + 2n - 5n - 1
\]

Simplify by combining like terms:

\[
10n^2 - 3n - 1
\]

So, the product is:

\[
\boxed{10n^2 - 3n - 1}
\]

Do you need more details or have any questions?

Here are 5 related questions:
1. How do you expand products of polynomials?
2. What is the FOIL method for multiplying binomials?
3. How do you combine like terms in an expression?
4. How would this process change if we were working with trinomials?
5. How does distributive property work in algebra?

**Tip:** Always check your sign when multiplying negative numbers to avoid small mistakes.

Learn how to multiply binomials using the distributive property (FOIL method) with the example (2n - 1)(5n + 1). Follow the step-by-step solution and simplify the expression to get 10n^2 - 3n - 1. This concept is key for understanding polynomial expansion, suitable for students in Grades 8-10.

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