Question

multiplyin polynomials and simplifying expressions
what is the product of $(3a + 2)(4a^2 - 2a + 9)$?
$12a^3 + 6a + 9$
$12a^3 + 2a^2 + 23a + 18$
$12a^3 - 6a^2 + 23a + 18$
$12a^3 - 2a + 18$

Explanation:

Step1: Apply distributive property (FOIL for polynomials)

Multiply \(3a\) by each term in \(4a^2 - 2a + 9\) and \(2\) by each term in \(4a^2 - 2a + 9\).
\(3a\times4a^2 = 12a^3\), \(3a\times(-2a)= -6a^2\), \(3a\times9 = 27a\)
\(2\times4a^2 = 8a^2\), \(2\times(-2a)= -4a\), \(2\times9 = 18\)

Step2: Combine like terms

Combine the results: \(12a^3 - 6a^2 + 27a + 8a^2 - 4a + 18\)
Combine \(a^2\) terms: \(-6a^2 + 8a^2 = 2a^2\)
Combine \(a\) terms: \(27a - 4a = 23a\)
So the expression becomes \(12a^3 + 2a^2 + 23a + 18\)

Answer:

\(12a^3 + 2a^2 + 23a + 18\) (corresponding to the option with this expression)