To expand \((a + 2b - 3c)^2\), we can apply the **square of a trinomial** formula:

\[
(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz
\]

Here, \( x = a \), \( y = 2b \), and \( z = -3c \). Let's break it down:

\[
(a + 2b - 3c)^2 = a^2 + (2b)^2 + (-3c)^2 + 2(a)(2b) + 2(a)(-3c) + 2(2b)(-3c)
\]

Now, calculate each term:

- \( a^2 = a^2 \)
- \( (2b)^2 = 4b^2 \)
- \( (-3c)^2 = 9c^2 \)
- \( 2(a)(2b) = 4ab \)
- \( 2(a)(-3c) = -6ac \)
- \( 2(2b)(-3c) = -12bc \)

Now combine everything:

\[
a^2 + 4b^2 + 9c^2 + 4ab - 6ac - 12bc
\]

So, the expansion of \((a + 2b - 3c)^2\) is:

\[
a^2 + 4b^2 + 9c^2 + 4ab - 6ac - 12bc
\]

Would you like a detailed explanation of any specific step?

Here are 5 related questions to expand on this:

1. How do you expand a binomial like \((a + b)^2\)?
2. What is the general formula for expanding \((x + y + z)^n\)?
3. Can you derive \((a - 2b + 3c)^2\) and compare the result?
4. How does the expansion change if one variable is squared, e.g., \((a^2 + 2b - 3c)^2\)?
5. How does the distributive property help in binomial and trinomial expansions?

**Tip:** Remember, terms involving multiple variables (like \( ab \) or \( ac \)) are known as **cross terms**, and they arise from multiplying different parts of the expression.

This page explains how to expand and simplify the trinomial (a + 2b - 3c)^2 using the square of a trinomial formula. The detailed solution covers polynomial expansion and algebraic concepts suitable for students in Grades 9-11.

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