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------------------------------------------------ Factor out the greatest common factor in the expression:

[tex]\[ 3x^4 - 21x^3 + 33x^2 \][/tex]

Sure! Let's factor out the greatest common factor (GCF) in the expression [tex]$3x^4 - 21x^3 + 33x^2$[/tex].

### Step-by-Step Solution:

1. Identify the GCF of the Coefficients:
- The coefficients of the terms are 3, 21, and 33.
- Find the GCF of these numbers:
- 3 can be divided by 3.
- 21 can be divided by 3.
- 33 can be divided by 3.
- So, the GCF of 3, 21, and 33 is 3.

2. Identify the GCF of the Variable Part:
- The expression has powers of [tex]$x$[/tex]: [tex]$x^4$[/tex], [tex]$x^3$[/tex], and [tex]$x^2$[/tex].
- The smallest power is [tex]$x^2$[/tex].
- Therefore, the GCF for the variable part is [tex]$x^2$[/tex].

3. Combine the GCFs:
- The overall GCF of the expression is [tex]$3x^2$[/tex].

4. Factor Out the GCF:
- Divide each term in the expression by the GCF [tex]$3x^2$[/tex]:
- [tex]$ \frac{3x^4}{3x^2} = x^2 $[/tex]
- [tex]$ \frac{-21x^3}{3x^2} = -7x $[/tex]
- [tex]$ \frac{33x^2}{3x^2} = 11 $[/tex]
- After factoring out the GCF, the expression inside is [tex]$x^2 - 7x + 11$[/tex].

5. Write the Final Expression:
- The factored expression is: [tex]$3x^2(x^2 - 7x + 11)$[/tex].

So, the factorization of the expression [tex]$3x^4 - 21x^3 + 33x^2$[/tex] is [tex]$3x^2(x^2 - 7x + 11)$[/tex].