Sure! Let's work through the expression step-by-step and simplify it.
Given the expression:
[tex]\[
(9 - x^2)(100 - y^2) - 120xy
\][/tex]
First, let's expand the product [tex]\((9 - x^2)(100 - y^2)\)[/tex]:
[tex]\[
(9 - x^2)(100 - y^2) = 9 \cdot 100 + 9 \cdot (-y^2) + (-x^2) \cdot 100 + (-x^2) \cdot (-y^2)
\][/tex]
This simplifies to:
[tex]\[
9 \cdot 100 + 9 \cdot (-y^2) + (-x^2) \cdot 100 + (-x^2) \cdot (-y^2) = 900 - 9y^2 - 100x^2 + x^2y^2
\][/tex]
So,
[tex]\[
(9 - x^2)(100 - y^2) = 900 - 9y^2 - 100x^2 + x^2y^2
\][/tex]
Now, we need to subtract [tex]\(120xy\)[/tex] from this result:
[tex]\[
900 - 9y^2 - 100x^2 + x^2y^2 - 120xy
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
-120xy + (9 - x^2)(100 - y^2)
\][/tex]
In its expanded form:
[tex]\[
900 - 9y^2 - 100x^2 + x^2y^2 - 120xy
\][/tex]
So, the final expression is:
[tex]\[
-120xy + (9 - x^2)(100 - y^2)
\][/tex]
This concludes the step-by-step simplification of the given expression.
