To factorize the given expression 12a^2b + 15ab^2, we need to find the common factors of the two terms.

Step 1: Factorize the coefficients.
The coefficient of the first term, 12, can be written as 3 * 4, and the coefficient of the second term, 15, can be written as 3 * 5.

Step 2: Factorize the variables.
The variable "a" is common to both terms, and it has an exponent of 2 in the first term and an exponent of 1 in the second term.
The variable "b" is also common to both terms, and it has an exponent of 1 in the first term and an exponent of 2 in the second term.

Step 3: Find the common factors.
The common factors in both terms are 3, a, and b.

Step 4: Write the factorized expression.
Using the common factors, we can rewrite the given expression as:
3ab * (4a + 5b)

So the correct answer is option 'A': 3ab(4a + 5b).

Explanation:
When we factorize an expression, we are essentially looking for the common factors in the terms of the expression. In this case, we can see that 3, a, and b are common factors in both terms.

By factoring out these common factors, we can simplify the expression and write it in a more compact form. In this case, we factor out 3ab from both terms of the expression, which leaves us with (4a + 5b) inside the parentheses.

The factorized form, 3ab(4a + 5b), is the simplest form of the expression and represents the original expression in a compact and organized way.

This process of factorization is useful in simplifying expressions, solving equations, and identifying patterns in mathematical problems. It allows us to break down complex expressions into simpler components and understand their underlying structure.