Alright, let's expand the expression [tex]\((-3x + 2y + z)^2\)[/tex] step by step.
Step 1: Recognize that squaring a binomial involves using the formula [tex]\((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the individual terms in the binomial. Here, [tex]\(a = -3x\)[/tex], [tex]\(b = 2y\)[/tex], and [tex]\(c = z\)[/tex].
Step 2: Square each term individually:
[tex]\[
(-3x)^2 = 9x^2
\][/tex]
[tex]\[
(2y)^2 = 4y^2
\][/tex]
[tex]\[
(z)^2 = z^2
\][/tex]
Step 3: Calculate the cross-product terms:
[tex]\[
2(-3x)(2y) = 2 \cdot (-3x) \cdot 2y = -12xy
\][/tex]
[tex]\[
2(-3x)(z) = 2 \cdot (-3x) \cdot z = -6xz
\][/tex]
[tex]\[
2(2y)(z) = 2 \cdot 2y \cdot z = 4yz
\][/tex]
Step 4: Combine all these terms to get the expanded form:
[tex]\[
9x^2 + 4y^2 + z^2 - 12xy - 6xz + 4yz
\][/tex]
Therefore, the expanded form of the expression [tex]\((-3x + 2y + z)^2\)[/tex] is:
[tex]\[
9x^2 + 4y^2 + z^2 - 12xy - 6xz + 4yz
\][/tex]
