(2a - 3b)(4a^2 + 6ab + 9b^2) =
recall that x^3 - y^3 = (x-y)(x^2+xy+y^2)
To multiply two binomials, we can use the distributive property. The distributive property states that for any numbers a, b, and c, the product of a multiplied by the sum of b and c is equal to the sum of the products of a multiplied by b and a multiplied by c.
In this case, we have the two binomials: (2a - 3b) and (4a^2 + 6ab + 9b^2). To multiply them, we will distribute each term from the first binomial to each term in the second binomial.
Let's break it down step by step:
First, multiply 2a with each term in the second binomial:
2a * 4a^2 = 8a^3
2a * 6ab = 12a^2b
2a * 9b^2 = 18ab^2
Next, multiply -3b with each term in the second binomial:
-3b * 4a^2 = -12a^2b
-3b * 6ab = -18ab^2
-3b * 9b^2 = -27b^3
Now, we can combine like terms by adding the products together:
8a^3 + 12a^2b + 18ab^2 - 12a^2b - 18ab^2 - 27b^3
When we combine like terms, we have:
8a^3 - 12a^2b - 12a^2b + 18ab^2 - 18ab^2 - 27b^3
Simplifying further:
8a^3 - 24a^2b + 0ab^2 - 27b^3
Finally, we can drop the 0-term and write the final answer as:
8a^3 - 24a^2b - 27b^3