Question

multiplying polynomials and simplifying expressions
what is the product of $a + 3$ and $-2a^2 + 15a + 6b^2$?
$\quad -2a^3 + 9a^2 + 45a + 24b^2$
$\quad -2a^3 + 21a^2 + 45a + 6ab^2 + 18b^2$
$\quad -2a^3 + 21a^2 + 45a + 24b^2$
$\quad -2a^3 + 9a^2 + 45a + 6ab^2 + 18b^2$

Explanation:

Step1: Apply distributive property

Multiply \(a\) and \(3\) with each term in \(-2a^{2}+15a + 6b^{2}\) separately.
First, multiply \(a\) by \(-2a^{2}+15a + 6b^{2}\):
\(a\times(-2a^{2})=-2a^{3}\), \(a\times15a = 15a^{2}\), \(a\times6b^{2}=6ab^{2}\)
Then, multiply \(3\) by \(-2a^{2}+15a + 6b^{2}\):
\(3\times(-2a^{2})=-6a^{2}\), \(3\times15a = 45a\), \(3\times6b^{2}=18b^{2}\)

Step2: Combine like terms

Now, combine the results from Step 1:
\(-2a^{3}+15a^{2}+6ab^{2}-6a^{2}+45a + 18b^{2}\)
Combine the \(a^{2}\) terms: \(15a^{2}-6a^{2}=9a^{2}\)
So the expression becomes: \(-2a^{3}+9a^{2}+45a + 6ab^{2}+18b^{2}\)

Answer:

\(-2a^{3}+9a^{2}+45a + 6ab^{2}+18b^{2}\) (the fourth option)