To factor the expression \( 45p^3 + 3p^2 + 30p + 2 \) completely, we can first look for a common factor in the first two terms and the last two terms.
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Group the terms:
\[ (45p^3 + 3p^2) + (30p + 2) \] -
Factor out the common factors in each group:
- In the first group \( 45p^3 + 3p^2 \), the common factor is \( 3p^2 \): \[ 3p^2(15p + 1) \]
- In the second group \( 30p + 2 \), the common factor is \( 2 \): \[ 2(15p + 1) \]
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Now we can re-write the expression with the factored groups: \[ 3p^2(15p + 1) + 2(15p + 1) \]
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Notice that \( (15p + 1) \) is a common factor: \[ (15p + 1)(3p^2 + 2) \]
Thus, the completely factored form of the expression \( 45p^3 + 3p^2 + 30p + 2 \) is: \[ (15p + 1)(3p^2 + 2) \]