Certainly! To factor the expression [tex]\( 4a^3 + 12a^2 + 8a \)[/tex], we will follow a detailed, step-by-step approach.
1. Identify the Greatest Common Factor (GCF):
First, we look for the greatest common factor among the terms. Each term in the expression [tex]\( 4a^3 + 12a^2 + 8a \)[/tex] has a common factor of [tex]\( 4a \)[/tex].
2. Factor out the GCF:
After identifying the GCF [tex]\( 4a \)[/tex], we factor it out from each term:
[tex]\[
4a^3 + 12a^2 + 8a = 4a(a^2) + 4a(3a) + 4a(2) = 4a(a^2 + 3a + 2)
\][/tex]
3. Factor the remaining quadratic expression:
Next, we focus on factoring the quadratic [tex]\( a^2 + 3a + 2 \)[/tex]. We need to find two numbers that multiply to [tex]\( 2 \)[/tex] (the constant term) and add to [tex]\( 3 \)[/tex] (the coefficient of the middle term). These numbers are [tex]\( 1 \)[/tex] and [tex]\( 2 \)[/tex]:
[tex]\[
a^2 + 3a + 2 = (a + 1)(a + 2)
\][/tex]
4. Combine the results:
Substituting back, we get:
[tex]\[
4a(a^2 + 3a + 2) = 4a(a + 1)(a + 2)
\][/tex]
Thus, the factored form of the expression [tex]\( 4a^3 + 12a^2 + 8a \)[/tex] is:
[tex]\[
\boxed{4a(a + 1)(a + 2)}
\][/tex]
