Certainly, let's examine each expression in detail and determine the number of terms in each one before organizing them from greatest to least by the number of terms.
1. Expression I: [tex]\(6x^2\)[/tex]
- This expression has a single term: [tex]\(6x^2\)[/tex].
- Number of terms: 1
2. Expression II: [tex]\(18x^2 + 5ab - 6y\)[/tex]
- This expression consists of three different terms: [tex]\(18x^2\)[/tex], [tex]\(5ab\)[/tex], and [tex]\(-6y\)[/tex].
- Number of terms: 3
3. Expression III: [tex]\(8a - 5\)[/tex]
- This expression contains two terms: [tex]\(8a\)[/tex] and [tex]\(-5\)[/tex].
- Number of terms: 2
4. Expression IV: [tex]\(4x^3 + 3x^2 - x - 4\)[/tex]
- This expression is composed of four terms: [tex]\(4x^3\)[/tex], [tex]\(3x^2\)[/tex], [tex]\(-x\)[/tex], and [tex]\(-4\)[/tex].
- Number of terms: 4
Now, let's organize these expressions from greatest to least by the number of terms:
- Expression IV has the greatest number of terms: 4 terms.
- Expression II follows with 3 terms.
- Expression III comes next with 2 terms.
- Expression I has the least number of terms: 1 term.
Therefore, the correct order from greatest to least by the number of terms is:
IV, II, III, I
So, the final answer is:
IV, II, III, I
