Certainly! Let's work through the expression step-by-step:
Given the expression:
[tex]\[
(9 - x^2)(100 - y^2) - 120xy
\][/tex]
Step 1: First, expand the product [tex]\((9 - x^2)(100 - y^2)\)[/tex].
Start by using the distributive property (FOIL method):
[tex]\[
(9 - x^2)(100 - y^2) = 9 \cdot 100 - 9 \cdot y^2 - x^2 \cdot 100 + x^2 \cdot y^2
\][/tex]
Let's multiply each term:
[tex]\[
9 \cdot 100 = 900
\][/tex]
[tex]\[
9 \cdot (-y^2) = -9y^2
\][/tex]
[tex]\[
(-x^2) \cdot 100 = -100x^2
\][/tex]
[tex]\[
(-x^2) \cdot (-y^2) = x^2 y^2
\][/tex]
Combining all these results:
[tex]\[
900 - 9y^2 - 100x^2 + x^2 y^2
\][/tex]
Step 2: Now subtract the [tex]\(120xy\)[/tex] term from the expanded expression:
[tex]\[
900 - 9y^2 - 100x^2 + x^2 y^2 - 120xy
\][/tex]
Therefore, the final expanded form of the expression [tex]\((9 - x^2)(100 - y^2) - 120xy\)[/tex] is:
[tex]\[
x^2 y^2 - 100x^2 - 120xy - 9y^2 + 900
\][/tex]
This is the completely expanded form of the given expression.
