To factor the polynomial \( 4x^2 - 6x + 8 \), let's start by looking for a common factor in all the terms:

1. **Factor out the GCD:** The greatest common factor (GCD) of the terms \(4x^2\), \(-6x\), and \(8\) is \(2\). Factoring out \(2\):

   \[
   4x^2 - 6x + 8 = 2(2x^2 - 3x + 4)
   \]

2. **Check for further factorization:** Now, we need to check if \(2x^2 - 3x + 4\) can be factored further. To do this, we look for two numbers that multiply to \(2 \times 4 = 8\) and add up to \(-3\). However, no such pair of numbers exists. Additionally, if you try using the quadratic formula, the discriminant \( b^2 - 4ac = (-3)^2 - 4(2)(4) = 9 - 32 = -23 \) is negative, which indicates that the expression cannot be factored over the real numbers.

Therefore, the fully factored form of the polynomial over the real numbers is:

\[
4x^2 - 6x + 8 = 2(2x^2 - 3x + 4)
\]

Would you like a more detailed breakdown, or do you have any other questions? Here are some related questions to consider:

1. How do you determine if a quadratic polynomial can be factored?
2. What are the steps for factoring polynomials with higher degrees?
3. How do you use the quadratic formula to find the roots of a polynomial?
4. What is the role of the discriminant in determining the nature of the roots?
5. How can we factor polynomials with complex numbers?

**Tip:** When factoring polynomials, always start by factoring out the greatest common factor (GCD) before trying other methods like grouping or the quadratic formula.

Learn how to factor the quadratic polynomial 4x^2 - 6x + 8 step-by-step. This explanation covers factoring out the GCD and using the discriminant to determine factorability over real numbers. Suitable for Grades 8-10 algebra students.

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