Answer :
We start with the expression
[tex]$$
\frac{2 x^6 - 18 x^4 + 2 x^2}{2 x^2}.
$$[/tex]
Step 1. Factor the numerator
Notice that every term in the numerator has a common factor of [tex]$2x^2$[/tex]. Factor it out:
[tex]$$
2x^6 - 18x^4 + 2x^2 = 2x^2 \left( x^4 - 9x^2 + 1 \right).
$$[/tex]
Step 2. Cancel the common factor
Now, substitute the factored numerator back into the expression:
[tex]$$
\frac{2x^2 \left( x^4 - 9x^2 + 1 \right)}{2x^2}.
$$[/tex]
Since [tex]$2x^2$[/tex] is present in both the numerator and the denominator (and assuming [tex]$x \neq 0$[/tex] so that we can cancel), we cancel this common factor:
[tex]$$
\frac{2x^2 \left( x^4 - 9x^2 + 1 \right)}{2x^2} = x^4 - 9x^2 + 1.
$$[/tex]
Step 3. Match with the provided choices
The simplified expression is
[tex]$$
x^4 - 9x^2 + 1,
$$[/tex]
which corresponds to the option:
D. [tex]$x^4 - 9 x^2 + 1.$[/tex]
Thus, the equivalent expression is
[tex]$$
\boxed{x^4 - 9x^2 + 1}.
$$[/tex]
[tex]$$
\frac{2 x^6 - 18 x^4 + 2 x^2}{2 x^2}.
$$[/tex]
Step 1. Factor the numerator
Notice that every term in the numerator has a common factor of [tex]$2x^2$[/tex]. Factor it out:
[tex]$$
2x^6 - 18x^4 + 2x^2 = 2x^2 \left( x^4 - 9x^2 + 1 \right).
$$[/tex]
Step 2. Cancel the common factor
Now, substitute the factored numerator back into the expression:
[tex]$$
\frac{2x^2 \left( x^4 - 9x^2 + 1 \right)}{2x^2}.
$$[/tex]
Since [tex]$2x^2$[/tex] is present in both the numerator and the denominator (and assuming [tex]$x \neq 0$[/tex] so that we can cancel), we cancel this common factor:
[tex]$$
\frac{2x^2 \left( x^4 - 9x^2 + 1 \right)}{2x^2} = x^4 - 9x^2 + 1.
$$[/tex]
Step 3. Match with the provided choices
The simplified expression is
[tex]$$
x^4 - 9x^2 + 1,
$$[/tex]
which corresponds to the option:
D. [tex]$x^4 - 9 x^2 + 1.$[/tex]
Thus, the equivalent expression is
[tex]$$
\boxed{x^4 - 9x^2 + 1}.
$$[/tex]
