Answer :

To factor out the greatest common factor (GCF) from the expression [tex]\(-3x^4 + 12x^3 - 21x^2\)[/tex], follow these steps:

1. Identify the GCF:
- Look at the coefficients: [tex]\(-3\)[/tex], [tex]\(12\)[/tex], and [tex]\(-21\)[/tex].
- The GCF of [tex]\(-3\)[/tex], [tex]\(12\)[/tex], and [tex]\(-21\)[/tex] is [tex]\(\-3\)[/tex].
- All terms also share a common variable [tex]\(x\)[/tex], and since the smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex] (in [tex]\(-21x^2\)[/tex]), the GCF for the variables is [tex]\(x^2\)[/tex].

2. Extract the GCF:
- The complete GCF is [tex]\(-3x^2\)[/tex].
- Factor [tex]\(-3x^2\)[/tex] out of each term:

[tex]\[
-3x^4 \div -3x^2 = x^2
\][/tex]

[tex]\[
12x^3 \div -3x^2 = -4x
\][/tex]

[tex]\[
-21x^2 \div -3x^2 = 7
\][/tex]

3. Write the factored expression:
- Combine the factored terms inside parentheses after the GCF:

[tex]\[
-3x^2(x^2 - 4x + 7)
\][/tex]

So, the expression [tex]\(-3x^4 + 12x^3 - 21x^2\)[/tex] factored by the greatest common factor is:

[tex]\[
-3x^2(x^2 - 4x + 7)
\][/tex]

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Rewritten by : Barada