To expand the expression \((2x + 1)(x + 9)\), we use the **distributive property** (also called FOIL method for binomials):

\[
(2x + 1)(x + 9) = 2x \cdot x + 2x \cdot 9 + 1 \cdot x + 1 \cdot 9
\]

Now, simplify each multiplication:

\[
2x \cdot x = 2x^2, \quad 2x \cdot 9 = 18x, \quad 1 \cdot x = x, \quad 1 \cdot 9 = 9
\]

Now, sum the terms:

\[
2x^2 + 18x + x + 9
\]

Combine like terms:

\[
2x^2 + 19x + 9
\]

Thus, the expanded form of \((2x + 1)(x + 9)\) is:

\[
\boxed{2x^2 + 19x + 9}
\]

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Do you want details or have any questions?  

Here are 5 related questions you might find helpful:  
1. How do you factor a quadratic expression like \(2x^2 + 19x + 9\)?  
2. What is the **FOIL method** and when should it be used?  
3. How do you solve quadratic equations derived from binomial multiplication?  
4. Can you provide the graph of \(y = 2x^2 + 19x + 9\)?  
5. How does the distributive property generalize to more terms, like trinomials?  

**Tip:** Always align like terms properly to avoid mistakes when expanding expressions.

Learn how to expand and simplify the binomial expression (2x + 1)(x + 9) using the distributive property and FOIL method. This problem covers key algebraic concepts such as binomial expansion and combining like terms. Ideal for students in Grades 7-9.

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