To determine which factored expression matches the given expression [tex]\( x^6 - y^4 + 2x^2 - 2y^2 \)[/tex], let's consider the options one by one.
The given expression is:
[tex]\[ x^6 - y^4 + 2x^2 - 2y^2 \][/tex]
We need to find which of the offered factored expressions corresponds to this.
### Option A:
[tex]\[ x^2\left(x^4 + 2\right) - y^2\left(y^2 + 2\right) \][/tex]
Let's expand this expression:
[tex]\[ x^2 \cdot x^4 + x^2 \cdot 2 - y^2 \cdot y^2 - y^2 \cdot 2 \][/tex]
[tex]\[ = x^6 + 2x^2 - y^4 - 2y^2 \][/tex]
This matches with the given expression.
Now, let's briefly consider the other options to ensure that none of them match as well.
### Option B:
[tex]\[ x^2\left(x^4 - 2\right) + y^2\left(y^2 - 2\right) \][/tex]
Let's expand this expression:
[tex]\[ x^2 \cdot x^4 - x^2 \cdot 2 + y^2 \cdot y^2 - y^2 \cdot 2 \][/tex]
[tex]\[ = x^6 - 2x^2 + y^4 - 2y^2 \][/tex]
This does not match the given expression.
### Option C:
[tex]\[ x^4\left(x^2 - 2\right) - y^2\left(y^4 + 2\right) \][/tex]
Let's expand this expression:
[tex]\[ x^4 \cdot x^2 - x^4 \cdot 2 - y^2 \cdot y^4 - y^2 \cdot 2 \][/tex]
[tex]\[ = x^6 - 2x^4 - y^6 - 2y^6 \][/tex]
This does not match the given expression.
### Option D:
[tex]\[ x^4\left(x^2 + 2\right) + y^2\left(y^4 - 2\right) \][/tex]
Let's expand this expression:
[tex]\[ x^4 \cdot x^2 + x^4 \cdot 2 + y^2 \cdot y^4 - y^2 \cdot 2 \][/tex]
[tex]\[ = x^6 + 2x^4 + y^6 - 2y^2 \][/tex]
This does not match the given expression.
Therefore, the correct factored expression is:
[tex]\[ \boxed{x^2\left(x^4 + 2\right) - y^2\left(y^2 + 2\right)} \][/tex]
