To expand the expression \((2x + 3y - z)^2\), we will use the formula for the square of a binomial, which states that \((a + b)^2 = a^2 + 2ab + b^2\). However, in this case, we have a trinomial, so we will need to apply the expansion carefully.
Step 1: Identify the Terms
- Let \(a = 2x\)
- Let \(b = 3y\)
- Let \(c = -z\)
Step 2: Apply the Formula
We can express \((a + b + c)^2\) as:
\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
\]
Step 3: Calculate Each Term
- \(a^2 = (2x)^2 = 4x^2\)
- \(b^2 = (3y)^2 = 9y^2\)
- \(c^2 = (-z)^2 = z^2\)
- \(2ab = 2(2x)(3y) = 12xy\)
- \(2ac = 2(2x)(-z) = -4xz\)
- \(2bc = 2(3y)(-z) = -6yz\)
Step 4: Combine All Terms
Putting all the calculated terms together, we get:
\[
(2x + 3y - z)^2 = 4x^2 + 9y^2 + z^2 + 12xy - 4xz - 6yz
\]
Final Result
Thus, the expanded form of \((2x + 3y - z)^2\) is:
\[
4x^2 + 9y^2 + z^2 + 12xy - 4xz - 6yz
\]
This step-by-step process allows you to clearly see how each component of the trinomial contributes to the final expanded expression.