To determine which expression represents a linear expression, we need to understand what characterizes a linear expression. A linear expression in [tex]\( x \)[/tex] is one where the highest power of [tex]\( x \)[/tex] is 1. In other words, it can be written in the form [tex]\( ax + b \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
Let's analyze each given expression one by one:
1. [tex]\(-17 x^4 - 18 x^3 + 19 x^2 - 20 x + 21\)[/tex]
- The terms in this expression include [tex]\( x^4 \)[/tex], [tex]\( x^3 \)[/tex], [tex]\( x^2 \)[/tex], [tex]\( x \)[/tex], and a constant.
- The highest power of [tex]\( x \)[/tex] is 4.
- Therefore, this is not a linear expression.
2. [tex]\(18 x^3 + 19 x^2 - 20 x + 21\)[/tex]
- The terms in this expression include [tex]\( x^3 \)[/tex], [tex]\( x^2 \)[/tex], [tex]\( x \)[/tex], and a constant.
- The highest power of [tex]\( x \)[/tex] is 3.
- Therefore, this is not a linear expression.
3. [tex]\(23 x^2 + 24 x - 25\)[/tex]
- The terms in this expression include [tex]\( x^2 \)[/tex], [tex]\( x \)[/tex], and a constant.
- The highest power of [tex]\( x \)[/tex] is 2.
- Therefore, this is not a linear expression.
4. [tex]\(4 x + 4\)[/tex]
- The terms in this expression include [tex]\( x \)[/tex] and a constant.
- The highest power of [tex]\( x \)[/tex] is 1.
- Therefore, this is a linear expression.
To sum it up, the only expression that represents a linear expression is:
[tex]\[ 4x + 4 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
