To find the highest common factor (HCF) of the given expressions [tex]\(\left(1-x^2\right)\left(1-y^2\right)+4xy\)[/tex] and [tex]\(1-2x+y-x^2 y + x^2\)[/tex], we need to follow these steps:
1. Expand the expressions:
First, let's expand [tex]\(\left(1 - x^2\right)\left(1 - y^2\right) + 4xy\)[/tex].
[tex]\[
(1 - x^2)(1 - y^2) + 4xy = (1 - x^2)(1 - y^2) + 4xy
\][/tex]
Expand [tex]\((1 - x^2)(1 - y^2)\)[/tex]:
[tex]\[
(1 - x^2)(1 - y^2) = 1 - y^2 - x^2 + x^2 y^2
\][/tex]
Now add [tex]\(4xy\)[/tex]:
[tex]\[
1 - y^2 - x^2 + x^2 y^2 + 4xy
\][/tex]
So the expanded form of the first expression is:
[tex]\[
1 - y^2 - x^2 + x^2 y^2 + 4xy
\][/tex]
2. Re-write the second expression:
The second expression is already in a simplified form:
[tex]\[
1 - 2x + y - x^2 y + x^2
\][/tex]
3. Find the Highest Common Factor (HCF):
To find the HCF of the expanded form of the first expression and the second expression:
The first expression can be written as:
[tex]\[
x^2 y^2 - x^2 - y^2 + 4xy + 1
\][/tex]
And the second expression is:
[tex]\[
1 - 2x + y - x^2 y + x^2
\][/tex]
After evaluating and simplifying the highest common factor of these two polynomials involving variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex], we derive:
[tex]\[
\boxed{x y - x + y + 1}
\][/tex]
Thus, the highest common factor of [tex]\(\left(1-x^2\right)\left(1-y^2\right)+4xy\)[/tex] and [tex]\(1-2x+y-x^2 y + x^2\)[/tex] is [tex]\(\boxed{xy - x + y + 1}\)[/tex].
