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When dealing with quadratic expressions, the first step in the factoring process is to find the Greatest Common Factor (GCF). The GCF is the largest factor that divides all the terms of the expression. This simplifies the expression and makes it easier to factor further.

Consider the quadratic expression from our example, \(2x^2 - 8x - 10\). The GCF for the coefficients (2, -8, and -10) is 2, because 2 is the largest number that divides all three coefficients without leaving a remainder. Therefore, we can factor out 2 from each term:

\[ 2(x^2 - 4x - 5) \]

Factoring out the GCF reduces the complexity of the quadratic equation, making it easier to tackle in the next steps.

Quadratic expressions are algebraic expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and the variable is raised to the power of 2. These expressions can often be factored into the product of two binomials. However, this is only if there exist two numbers that multiply to give the constant term \(c\) and add to give the coefficient \(b\).

Let's focus on the simplified expression inside the parentheses from our example: \(x^2 - 4x - 5\)

The goal here is to find two numbers that multiply to \(-5\) (the constant term) and add to \(-4\) (the coefficient of the x-term). It helps to list out pairs of factors of \(-5\). In this case:

• \(-5 \cdot 1 = -5\)
• \(-5 + 1 = -4\)

Hence, \(-5\) and \(1\) are the factors we need.

Now we can rewrite the quadratic expression as a product of two binomials: \((x - 5)(x + 1)\).

The factoring process involves breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. Here’s a simple 6-step guide to factor quadratics effectively:

• Identify the quadratic expression to be factored.
• Find and factor out the Greatest Common Factor (GCF).
• Focus on the simplified quadratic expression inside the parentheses.
• Find two numbers that multiply to the constant term and add to the coefficient of the x-term.
• Rewrite the quadratic expression as the product of two binomials.
• Combine the factors with the GCF initially factored out.

Let’s apply these steps to our example:
\(2x^2 - 8x - 10\)

After factoring out the GCF of 2, we simplify the quadratic to \(x^2 - 4x - 5\).

Since \(-5\) and \(1\) are the factors needed, rewrite it as \((x - 5)(x + 1)\).

Putting it all together, we get:

\[2(x - 5)(x + 1)\]

And that's the completely factored form!