Answer :

GinnyAnswer
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]

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