To expand the expression \((2x + 3y - 5z)^2\), we will use the formula for the square of a binomial \((a + b + c)^2\):

\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
\]

Here, \(a = 2x\), \(b = 3y\), and \(c = -5z\). Let's apply the formula:

\[
(2x + 3y - 5z)^2 = (2x)^2 + (3y)^2 + (-5z)^2 + 2(2x)(3y) + 2(2x)(-5z) + 2(3y)(-5z)
\]

Now, calculate each term:

\[
(2x)^2 = 4x^2
\]
\[
(3y)^2 = 9y^2
\]
\[
(-5z)^2 = 25z^2
\]
\[
2(2x)(3y) = 12xy
\]
\[
2(2x)(-5z) = -20xz
\]
\[
2(3y)(-5z) = -30yz
\]

Combine all these terms:

\[
(2x + 3y - 5z)^2 = 4x^2 + 9y^2 + 25z^2 + 12xy - 20xz - 30yz
\]

So, the expanded form of \((2x + 3y - 5z)^2\) is:

\[
4x^2 + 9y^2 + 25z^2 + 12xy - 20xz - 30yz
\]

Would you like more details or have any questions? Here are five questions you might consider asking next:

1. How do you expand \((3a - 4b + 2c)^2\)?
2. Can you explain the binomial theorem in more detail?
3. What is the process for factoring a polynomial?
4. How do you find the derivative of a function involving multiple variables?
5. Can you demonstrate how to solve a system of linear equations?

**Tip:** When expanding polynomials, keep track of all terms and their signs carefully to avoid mistakes.

Learn how to expand the expression (2x + 3y - 5z)^2 step-by-step with a detailed solution. This problem covers key algebraic concepts, including the binomial theorem and polynomial expansions. Suitable for high school students.

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