Question: Expand the product $ (2p + q)(p - 3q)(p + q) $. - Coaching Toolbox

Understanding the Context

In a digital landscape increasingly shaped by connections, variables and patterns, solving complex expressions is more than a math exercise—it’s a foundational skill. The expansion of $ (2p + q)(p - 3q)(p + q) $ is quietly gaining traction among students, educators, and professionals exploring algebra’s practical impact. With growing emphasis on analytical thinking and computational literacy, efficiently expanding this trinomial product serves as a gateway to understanding real-world relationships in fields ranging from economics to data science.

Though few recognize it by name, this expansion underpins how systems model interactions—whether predicting market dynamics, analyzing educational trends, or optimizing resource allocation. Understanding how to simplify such expressions fosters a deeper grasp of algebraic structures that power more advanced problem-solving.

Why This Question Is Standing Out in the US Context

The rise in interest stems from broader educational and professional trends across the United States. Students increasingly rely on structured problem-solving frameworks to tackle STEM subjects, while professionals in data analysis and finance seek clear, repeatable methods for modeling variable interactions. Educational resources emphasize procedural fluency now more than ever, making clear explanations of expressions like $ (2p + q)(p - 3q)(p + q) $ both timely and valuable.

Key Insights

The move reflects a national shift toward computational thinking—one where breaking down and expanding expressions isn’t just academic, but a practical habit enabling clearer insights into complex systems.

How to Expand $ (2p + q)(p - 3q)(p + q) $: A Clear, Step-by-Step Process

Expanding this product involves careful application of the distributive property, maintaining clarity and accuracy. Start by recognizing that the expression multiplies three binomials—each pair must be multiplied in sequence.

First, multiply $ (2p + q) $ and $ (p + q) $ using distributive multiplication:
$ (2p + q)(p + q) = 2p(p + q) + q(p + q) = 2p^2 + 2pq + pq + q^2 = 2p^2 + 3pq + q^2 $.

Now multiply that result by $ (p - 3q) $, applying distributive property again:
$ (2p^2 + 3pq + q^2)(p - 3q) = 2p^2(p - 3q) + 3pq(p - 3q) + q^2(p - 3q) $.

Final Thoughts

Breaking each term:

• $ 2p^2(p) = 2p^3 $, $ 2p^2(-3q) = -6p^2q $
• $ 3pq(p) = 3p^2q $, $ 3pq(-3q) = -9pq^2 $
• $ q^2(p) = pq^2 $, $ q^2(-3q) = -3q^3 $

Combine all:
$ 2p^3 - 6p^2q + 3p^2q - 9pq^2 + pq^2 - 3q^3 $

Simplify like terms:
$ 2p^3 - 3p^2q - 8pq^2 - 3q^3 $

This expanded form