Let's solve the given expression [tex]\((4xy - 3z)^2\)[/tex] and identify the equivalent expression and its type of special product.
1. Expand [tex]\((4xy - 3z)^2\)[/tex]:
[tex]\[
(4xy - 3z)^2 = (4xy - 3z)(4xy - 3z)
\][/tex]
2. Apply the distributive property (FOIL method) to expand the expression:
[tex]\[
(4xy - 3z)(4xy - 3z) = 4xy \cdot 4xy + 4xy \cdot (-3z) + (-3z) \cdot 4xy + (-3z) \cdot (-3z)
\][/tex]
3. Simplify each term:
- First term: [tex]\(4xy \cdot 4xy = 16x^2y^2\)[/tex]
- Second term: [tex]\(4xy \cdot (-3z) = -12xyz\)[/tex]
- Third term: [tex]\((-3z) \cdot 4xy = -12xyz\)[/tex]
- Fourth term: [tex]\((-3z) \cdot (-3z) = 9z^2\)[/tex]
4. Combine like terms:
- Combine the second and third terms:
[tex]\[
-12xyz - 12xyz = -24xyz
\][/tex]
- Now add all terms together:
[tex]\[
16x^2y^2 - 24xyz + 9z^2
\][/tex]
The expanded form of [tex]\((4xy - 3z)^2\)[/tex] is:
[tex]\[
16x^2y^2 - 24xyz + 9z^2
\][/tex]
5. Identify the type of special product:
- The expression obtained is [tex]\(16x^2y^2 - 24xyz + 9z^2\)[/tex], which is a perfect square trinomial, as it conforms to the form [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
Therefore, the expression equivalent to [tex]\((4xy - 3z)^2\)[/tex] is [tex]\(16x^2y^2 - 24xyz + 9z^2\)[/tex], and it is a perfect square trinomial.
So, the correct answer is:
[tex]\[
16x^2y^2 - 24xy z + 9z^2, \text{ a perfect square trinomial}
\][/tex]
The corresponding option is:
[tex]\[
4. \, 16x^2y^2 - 24xy z + 9z^2, \text{ a perfect square trinomial}
\][/tex]
