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Factor `8x^{2}+16x-10`completely. Which of the following expressions is a factor?

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1 answer

To factor the quadratic expression \(8x^2 + 16x - 10\) completely, we can follow these steps:

  1. Look for a common factor: The coefficients of the terms \(8\), \(16\), and \(-10\) have a common factor of \(2\). We can factor \(2\) out of the expression: \[ 8x^2 + 16x - 10 = 2(4x^2 + 8x - 5) \]

  2. Factor the quadratic expression \(4x^2 + 8x - 5\): We are looking for two numbers that multiply to \(4 \times -5 = -20\) and add to \(8\).

    After checking pairs of factors, we find: The numbers that satisfy these conditions are \(10\) and \(-2\) because: \[ 10 + (-2) = 8 \] and \[ 10 \cdot (-2) = -20 \]

  3. Rewrite the middle term: Rewrite \(8x\) using the numbers found: \[ 4x^2 + 10x - 2x - 5 \]

  4. Group the terms: We can now group the terms: \[ (4x^2 + 10x) + (-2x - 5) \]

  5. Factor by grouping: Now factor out the common factors in each group: \[ 2x(2x + 5) - 1(2x + 5) \]

  6. Factor out the common binomial: Now, factor out the common binomial factor \((2x + 5)\): \[ (2x + 5)(2x - 1) \]

  7. Combine with the common factor: Adding back the factor of \(2\) we mentioned initially, we have: \[ 8x^2 + 16x - 10 = 2(2x + 5)(2x - 1) \]

Now, the completely factored form of \(8x^2 + 16x - 10\) is: \[ 2(2x + 5)(2x - 1) \]

Therefore, one of the factors of the original expression \(8x^2 + 16x - 10\) is \((2x + 5)\) or \((2x - 1)\), along with the common factor \(2\).

If your task was to find just one of the factors, choosing either \((2x + 5)\) or \((2x - 1)\) would be correct.