Factor the greatest common factor: 12a3b + 8a2b2 − 20ab3. 4ab(3a + 2ab − 5b2) 4ab(3a2 + 2ab − 5b2) 4a2b(3a3b − 2a2b2 − 5ab3) 4a2b(3a2 + 2ab − 5)

Step 1: Break Down the Problem

We need to factor the expression $12a^3b + 8a^2b^2 - 20ab^3$ by finding the greatest common factor (GCF) among the terms.

Step 2: Relevant Concepts

1. Identify the GCF: The GCF can be found by looking for the highest degree of $a$ and $b$ in each term.
2. Factor out the GCF: This involves rewriting the expression so that the GCF is multiplied by the remaining expression.

Step 3: Analysis and Detail

1. The terms to factor are:

   • $12a^3b$
   • $8a^2b^2$
   • $-20ab^3$

2. Finding the GCF:

   • For the coefficients (12, 8, -20), the GCF is $4$.
   • For $a$, the lowest exponent is $a^1$ (from $-20ab^3$).
   • For $b$, the lowest exponent is $b^1$ (from $12a^3b$).
   • Thus, the GCF is $4ab$.

3. Factoring out the GCF: $12a^3b + 8a^2b^2 - 20ab^3 = 4ab(3a^2 + 2ab - 5b^2)$

Step 4: Verify and Summarize

1. Verification:
   • Distributing $4ab$: $4ab \cdot 3a^2 = 12a^3b$ $4ab \cdot 2ab = 8a^2b^2$ $4ab \cdot (-5b^2) = -20ab^3$
   • The verification shows that they match the original expression.

Final Answer

The factored expression is: $4ab(3a^2 + 2ab - 5b^2)$