Question

what is the product of a+3 and -2a²+15a+6b²?
-2a³+9a²+45a+24b²
-2a³+21a²+45a+24b²
-2a³+9a²+45a+6ab²+18b²
-2a³+21a²+45a+6ab²+18b²

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 Step1:
For the \(a^{3}\) term: \(-2a^{3}\) (only one term)
For the \(a^{2}\) terms: \(15a^{2}-6a^{2}=9a^{2}\)
For the \(a\) terms: \(45a\) (only one term)
For the \(ab^{2}\) term: \(6ab^{2}\) (only one term)
For the \(b^{2}\) term: \(18b^{2}\) (only one term)

Putting it all together: \(-2a^{3}+9a^{2}+45a + 6ab^{2}+18b^{2}\)

Answer:

-2a³+9a²+45a+6ab²+18b² (the third option)