To factor the polynomial \( 4n^3 - 12n^2 + 3n - 9 \), we can begin by grouping the terms.
First, we can group the first two terms and the last two terms:
\[ (4n^3 - 12n^2) + (3n - 9) \]
Now, we can factor out the greatest common factor from each group:
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For \( 4n^3 - 12n^2 \), the GCF is \( 4n^2 \): \[ 4n^2(n - 3) \]
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For \( 3n - 9 \), the GCF is \( 3 \): \[ 3(n - 3) \]
Now, substitute these factored forms back into the expression:
\[ 4n^2(n - 3) + 3(n - 3) \]
Notice that \( (n - 3) \) is a common factor:
\[ (n - 3)(4n^2 + 3) \]
Thus, the factored form of the polynomial \( 4n^3 - 12n^2 + 3n - 9 \) is:
\[ (n - 3)(4n^2 + 3) \]
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