Verified Answer :

  • First, factor out the greatest common divisor (GCD) of 8 from the expression, which yields \(8(2g^2 + g - 10)\).
  • Next, factor the quadratic trinomial \(2g^2 + g - 10\) by finding two integers whose product is \(-10\) and whose contribution matches the linear coefficient, resulting in \((2g + 5)(g - 2)\).
  • Combine these results to arrive at the final factored expression: \(\boxed{8(2g + 5)(g - 2)}\).

Explanation

  1. Factor out the Greatest Common Divisor

We are given the quadratic expression \(16g^2 + 8g - 80\). To factor it completely, we first look for a common factor among all coefficients: 16, 8, and -80. Notice that all these numbers are divisible by 8. We factor out the greatest common divisor:
\[16g^2 + 8g - 80 = 8(2g^2 + g - 10).\]

  1. Factor the Quadratic Trinomial

Now we need to factor the quadratic trinomial \(2g^2 + g - 10\). We look for factors of the form \((2g + m)(g + n) = 2g^2 + 2ng + mg + mn = 2g^2 + (2n + m)g + mn\).
Comparing this to \(2g^2 + g - 10\), we need:

  1. \(mn = -10\)
  2. \(2n + m = 1\)

Testing factors of -10:

  • If \(n = -2\) and \(m = 5\), then \(mn = -10\) and \(2n + m = 2(-2) + 5 = -4 + 5 = 1\).

This fits perfectly! Thus, \(2g^2 + g - 10 = (2g + 5)(g - 2)\).

  1. Combine the Results

Finally, we combine the common factor 8 with the factored trinomial to get the complete factorization:
\[16g^2 + 8g - 80 = 8(2g + 5)(g - 2).\]
This is the complete factored form of the expression.

Examples

Factoring polynomials is a fundamental tool in algebra, much like breaking down a complex machine into its basic, working parts. For example, if you are designing a robotic arm's movement path governed by the equation \(16g^2 + 8g - 80 = 0\) (where \(g\) could represent time or position), factoring it allows you to quickly solve for the exact moments or locations (\(g = 2\) or \(g = -2.5\)) where the arm reaches its target. Understanding these components makes analyzing systems in engineering, physics, and computer science much simpler.

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