To solve the given equation [tex]\((xyz)^2 - 1 = \square \cdot (xyz - 1)\)[/tex], we start by utilizing the algebraic identity known as the difference of squares.
The difference of squares states that:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Let's rewrite our given expression in a way that can exploit this identity. We'll set [tex]\(a = xyz\)[/tex] and [tex]\(b = 1\)[/tex]. We have:
[tex]\[
(xyz)^2 - 1^2 = (xyz - 1)(xyz + 1)
\][/tex]
Now, compare the left-hand side of the given equation with this identity. The left-hand side [tex]\((xyz)^2 - 1\)[/tex] matches [tex]\(a^2 - b^2\)[/tex], where [tex]\(a = xyz\)[/tex] and [tex]\(b = 1\)[/tex].
According to the difference of squares identity:
[tex]\[
(xyz)^2 - 1 = (xyz - 1)(xyz + 1)
\][/tex]
This shows that the expression on the right-hand side, [tex]\(\square\)[/tex], must be [tex]\(xyz + 1\)[/tex] to match the structure of the difference of squares identity.
Therefore, the completed equation looks like:
[tex]\[
(xyz)^2 - 1 = (xyz + 1) \cdot (xyz - 1)
\][/tex]
Hence, the expression that correctly completes the provided equation is:
[tex]\[
(xyz + 1)
\][/tex]
