To factorize the expression $25a^2 - 4b^2 + 20ac + 8bc$, we can follow these steps:

Step 1: Group terms in a strategic way

$25a^2 + 20ac - 4b^2 + 8bc$

Step 2: Factorize by grouping

Group the terms as follows: $(25a^2 + 20ac) - (4b^2 - 8bc)$

Step 3: Factorize each group

Factor out the common factors in each group: $5a(5a + 4c) - 4b(b - 2c)$

Step 4: Combine using common binomial factor

Notice that $(5a + 4c)$ and $(b - 2c)$ are the factors. Thus, rewrite the expression as: $(5a - 4b)(5a + 4c)$

Final Answer:

$(5a - 4b)(5a + 4c)$

Would you like a step-by-step breakdown of any specific part? Here are 5 related questions for further exploration:

1. How do you identify common factors in a polynomial?
2. Can you explain the difference between grouping and substitution in factorization?
3. How do you verify if a factorization is correct?
4. What are some common strategies for factorizing four-term polynomials?
5. How does recognizing patterns like the difference of squares help in factorization?

Tip: Always double-check your factorization by expanding the factors to see if you retrieve the original expression.