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------------------------------------------------ Find the greatest common factor of these three expressions:

[tex]\[ 21x^2, \, 9x^4, \, \text{and} \, 45. \][/tex]

To find the greatest common factor (GCF) of the expressions [tex]\(21x^2\)[/tex], [tex]\(9x^4\)[/tex], and [tex]\(45\)[/tex], we'll follow these steps:

1. Identify the numerical coefficients:
- The coefficient of [tex]\(21x^2\)[/tex] is 21.
- The coefficient of [tex]\(9x^4\)[/tex] is 9.
- The coefficient of 45 is 45.

2. Find the GCF of the numerical coefficients:
- First, list the factors of each number:
- Factors of 21: 1, 3, 7, 21
- Factors of 9: 1, 3, 9
- Factors of 45: 1, 3, 5, 9, 15, 45
- Identify the common factors:
- The common factors of 21, 9, and 45 are: 1 and 3.
- The greatest of these common factors is 3.

3. Consider any variables involved:
- The expressions involve [tex]\(x^2\)[/tex] and [tex]\(x^4\)[/tex], but since 45 has no variable, we only focus on the coefficients for determining the GCF.

Therefore, the greatest common factor of the expressions [tex]\(21x^2\)[/tex], [tex]\(9x^4\)[/tex], and [tex]\(45\)[/tex] is 3.