To factorize the expression $2a^2b - 8b^2c - 4abc + 4ab^2$, let's proceed step by step:

Expression:

$2a^2b - 8b^2c - 4abc + 4ab^2$

Step 1: Group terms that have common factors.

We will group the terms as follows: $(2a^2b - 4abc) + (-8b^2c + 4ab^2)$

Step 2: Factor out the common factors in each group.

For the first group $(2a^2b - 4abc)$, the common factor is $2ab$: $2ab(a - 2c)$ For the second group $(-8b^2c + 4ab^2)$, the common factor is $4b^2$: $4b^2(-2c + a)$ which simplifies to: $4b^2(a - 2c)$

Step 3: Factor out the common binomial factor.

Now, we can factor out the common binomial factor $(a - 2c)$: $(2ab + 4b^2)(a - 2c)$

Step 4: Simplify the remaining factor.

Notice that $2ab + 4b^2$ has a common factor of $2b$, so we factor it out: $2b(a + 2b)(a - 2c)$

Final Factorized Form:

The fully factorized expression is: $2b(a + 2b)(a - 2c)$

Would you like more details on any of these steps or have any questions?

Here are some related practice problems:

1. Factorize $3x^2y - 6xy^2 + 9xyz$.
2. Factorize $4m^2n - 8mn^2 + 12mn$.
3. Factorize $6pq - 9pr + 12qr$.
4. Factorize $5x^2y - 10xy + 15xy^2$.
5. Factorize $7a^2b - 14ab^2 + 21abc$.

Tip: Always look for the greatest common factor (GCF) when factorizing algebraic expressions to simplify the process.