Sure! To find the area of the rectangle, we need to multiply the length and the width and then simplify the resulting expression.
### Given:
Length: [tex]\( x^2 - 2 \)[/tex]
Width: [tex]\( 2x^2 - x + 2 \)[/tex]
### Step-by-Step Multiplication:
1. Distribute each term in the length to each term in the width:
[tex]\[
(x^2 - 2)(2x^2 - x + 2)
\][/tex]
2. Expand by distributing [tex]\(x^2\)[/tex] to each term in the width:
[tex]\[
x^2 \cdot 2x^2 + x^2 \cdot (-x) + x^2 \cdot 2
\][/tex]
This results in:
[tex]\[
2x^4 - x^3 + 2x^2
\][/tex]
3. Expand by distributing [tex]\(-2\)[/tex] to each term in the width:
[tex]\[
-2 \cdot 2x^2 - 2 \cdot (-x) - 2 \cdot 2
\][/tex]
This results in:
[tex]\[
-4x^2 + 2x - 4
\][/tex]
4. Combine all the expanded terms together:
[tex]\[
2x^4 - x^3 + 2x^2 - 4x^2 + 2x - 4
\][/tex]
5. Combine like terms:
First, simplify [tex]\(2x^2 - 4x^2\)[/tex]:
[tex]\[
2x^2 - 4x^2 = -2x^2
\][/tex]
Now, combine all the terms:
[tex]\[
2x^4 - x^3 - 2x^2 + 2x - 4
\][/tex]
### Result:
The area of the rectangle, expressed as a trinomial in its simplest form, is:
[tex]\[
\boxed{2x^4 - x^3 - 2x^2 + 2x - 4}
\][/tex]
