To find the product of the expressions \((4j + 2)(4j^2 + j - 4)\), we will use the distributive property (also known as the FOIL method when dealing with binomials).
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Distribute \(4j\) to each term in the second polynomial: \[ 4j \cdot 4j^2 = 16j^3 \] \[ 4j \cdot j = 4j^2 \] \[ 4j \cdot (-4) = -16j \]
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Now distribute \(2\) to each term in the second polynomial: \[ 2 \cdot 4j^2 = 8j^2 \] \[ 2 \cdot j = 2j \] \[ 2 \cdot (-4) = -8 \]
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Now combine all the resulting terms: \[ 16j^3 + 4j^2 - 16j + 8j^2 + 2j - 8 \]
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Combine like terms:
- The \(j^3\) term: \(16j^3\)
- The \(j^2\) terms: \(4j^2 + 8j^2 = 12j^2\)
- The \(j\) terms: \(-16j + 2j = -14j\)
- The constant term: \(-8\)
So, putting it all together, we get: \[ 16j^3 + 12j^2 - 14j - 8 \]
Thus, the simplified product is: \[ \boxed{16j^3 + 12j^2 - 14j - 8} \]
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