The given quadratic expression is:

$3x^2 - 2x - 8$

To factor this expression:

1. Multiply the coefficient of $x^2$ (which is 3) by the constant term (which is -8): $3 \times -8 = -24$
2. Find two numbers that multiply to give -24 and add to give the middle coefficient (-2):
   • The numbers that work are 4 and -6 because: $4 \times -6 = -24 \quad \text{and} \quad 4 + (-6) = -2$
3. Rewrite the middle term using these two numbers: $3x^2 + 4x - 6x - 8$
4. Factor by grouping:
   • Group the terms as: $(3x^2 + 4x) - (6x + 8)$
   • Factor each group: $x(3x + 4) - 2(3x + 4)$
5. Factor out the common binomial factor (3x + 4): $(x - 2)(3x + 4)$

Final factored form:

$(3x + 4)(x - 2)$

Would you like further explanation on any part of this factoring process?

Here are some follow-up questions to deepen understanding:

1. Can you explain why we used the numbers 4 and -6?
2. What is the purpose of grouping the terms in step 4?
3. Can all quadratic expressions be factored like this?
4. How can we verify the factorization is correct?
5. What happens if we couldn't find two numbers that multiply to the given product and add to the middle term?

Tip: When factoring quadratics, always check for a common factor first before applying more advanced techniques like grouping.