To factor the expression \( 6p^4 + 21p^3 + 9p^2 \), we start by looking for a common factor in all the terms.
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Find the Greatest Common Factor (GCF):
- The coefficients are 6, 21, and 9. The GCF of these numbers is 3.
- The lowest power of \( p \) present in all terms is \( p^2 \).
Therefore, the GCF of the entire expression is \( 3p^2 \).
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Factor out the GCF: \[ 6p^4 + 21p^3 + 9p^2 = 3p^2(2p^2 + 7p + 3) \]
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Factor the quadratic \( 2p^2 + 7p + 3 \): We look for two numbers that multiply to \( 2 \cdot 3 = 6 \) (the product of the coefficient of \( p^2 \) and the constant) and add to \( 7 \) (the coefficient of \( p \)). The numbers \( 6 \) and \( 1 \) fit this requirement.
Using these numbers, we can rewrite the middle term: \[ 2p^2 + 6p + 1p + 3 \]
Then we group the terms: \[ = (2p^2 + 6p) + (1p + 3) \] \[ = 2p(p + 3) + 1(p + 3) \]
Now we can factor out the common binomial \( (p + 3) \): \[ = (2p + 1)(p + 3) \]
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Combine everything together: Thus, the completely factored form of the original expression is: \[ 6p^4 + 21p^3 + 9p^2 = 3p^2(2p + 1)(p + 3) \]
So, the final answer is: \[ \boxed{3p^2(2p + 1)(p + 3)} \]