To determine the correct step that Martin missed, we need to analyze how the equation \(3x^2 + 2x - 8 = 0\) is factored.
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Factoring the Equation: The original equation is \(3x^2 + 2x - 8 = 0\). Its factors can be found by looking for pairs of terms that can be grouped.
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Group the Terms: The equation can potentially be grouped based on the coefficients of \(x^2\) and \(x\).
A common method is to use the technique of factoring by grouping or finding a common factor.
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Finding GCF: The greatest common factor (GCF) for each pair of grouped terms would need to be identified.
- One possible factorization would group the first two and the last term: \[ 3x^2 + 6x - 4x - 8 = 0 \] This can be grouped as: \[ (3x^2 + 6x) + (-4x - 8) = 0 \] The GCF of \(3x^2 + 6x\) is \(3x\) and for \(-4x - 8\) it is \(-4\).
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Setting up as an Equation: From this we might have: \[ 3x(x + 2) - 4(x + 2) = 0 \] In this case, \(x + 2\) is a common binomial factor. This can be further factored out.
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Final Factorization: Once factoring \(x + 2\) yields: \[ (3x - 4)(x + 2) = 0 \]
Now let's go through the provided options based on our findings:
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Option 1: \( x(3x + 6) + 4(x + 2) = 0 \) is incorrect as it doesn't reflect the work correctly and possibly has an error in signs.
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Option 2: \( 3x(x + 2) - 4(x - 2) = 0 \) is also incorrect since the second term should be \( -4(x + 2) \).
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Option 3: \( 3x(x + 2) + 4(x + 2) = 0 \) includes the common term added instead of subtracted, which doesn't represent correct factoring either.
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Option 4: \( 3x(x + 2) - 4(x + 2) = 0 \) is correct as it shows both parts factored properly.
Thus, the correct missing part of Martin's work is:
Option 4: \(3x(x + 2) - 4(x + 2) = 0\).
This aligns correctly with the factoring steps we went through above.
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