To fully factorize $3x^2 - 2xy - 8y^2$, follow these steps:

1. Multiply the coefficient of $x^2$ (3) by the constant term (-8): $3 \times -8 = -24$.
2. Find two numbers that multiply to -24 and add to the middle term's coefficient (-2): These numbers are -6 and 4.
3. Rewrite the middle term using -6 and 4: $3x^2 - 6xy + 4xy - 8y^2$
4. Group terms in pairs: $(3x^2 - 6xy) + (4xy - 8y^2)$
5. Factor out the greatest common factor (GCF) from each group: $3x(x - 2y) + 4y(x - 2y)$
6. Factor out the common binomial factor $(x - 2y)$: $(3x + 4y)(x - 2y)$

The fully factorized form is: $(3x + 4y)(x - 2y)$

Would you like more details or have any questions? Here are related problems for practice:

1. Factorize $6x^2 + 11xy + 4y^2$.
2. Solve $3x^2 - 2xy - 8y^2 = 0$.
3. Expand $(3x + 4y)(x - 2y)$ to verify.
4. Factorize $x^2 - 9$.
5. Factorize $4x^2 - 25y^2$.

Tip: Always check your factorization by expanding it back to the original expression!