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To factor the quadratic expression \(3n^2 - 8n + 4\), you can use the method of factoring by grouping or the quadratic formula to find its roots. Here’s one efficient way to do it: 1. **Find two numbers that multiply to \(a \cdot c\) (the coefficient of \(n^2\) times the constant term) and add up to \(b\) (the coefficient of \(n\)):** For the quadratic expression \(3n^2 - 8n + 4\): - \(a = 3\) - \(b = -8\) - \(c = 4\) We are looking for two numbers that multiply to \(3 \cdot 4 = 12\) and add up to \(-8\). 2. The two numbers that work are \(-6\) and \(-2\) because: - \(-6 \times -2 = 12\) - \(-6 + (-2) = -8\) 3. **Rewrite the middle term \(-8n\) using \(-6n\) and \(-2n\):** \[ 3n^2 - 6n - 2n + 4 \] 4. **Group the terms and factor each group:** \[ (3n^2 - 6n) - (2n - 4) \] Factor out the greatest common factor (GCF) from each group: \[ 3n(n - 2) - 2(n - 2) \] 5. **Factor out the common binomial \((n - 2)\):** \[ (3n - 2)(n - 2) \] So, the factored form of \(3n^2 - 8n + 4\) is: \[ (3n - 2)(n - 2) \] You can double-check your work by expanding \((3n - 2)(n - 2)\) to ensure it yields the original expression \(3n^2 - 8n + 4\).

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