To factor the expression \(-8x^3 - 2x^2 - 12x - 3\) by grouping, we can follow these steps:
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Group the terms: We can group as follows: \((-8x^3 - 2x^2)\) and \((-12x - 3)\).
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Factor out the common terms in each group:
- From the first group \(-8x^3 - 2x^2\), we can factor out \(-2x^2\): \[ -2x^2(4x + 1) \]
- From the second group \(-12x - 3\), we can factor out \(-3\): \[ -3(4x + 1) \]
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Combine the factored groups: Now we have: \[ -2x^2(4x + 1) - 3(4x + 1) \]
Notice that both terms contain the common factor \((4x + 1)\), which we can factor out: \[ (4x + 1)(-2x^2 - 3) \]
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Rearranging: We often prefer to write the polynomial in descending order. The expression can be rewritten to show it in a standard form: \[ (4x + 1)(-2x^2 - 3) \] Since we want a specific form where all terms are grouped, we multiply out the first term if necessary.
From these steps, the resulting expression after grouping and factoring is:
\[ (4x + 1)(-2x^2 - 3) \]
Comparing this to the answer choices, we can see: \[ (-2x^2 - 3)(4x + 1) \]
Thus, the correct answer from the choices is:
\[ \textbf{(–2x^2 – 3)(4x + 1)} \]