To factor the expression \(21a^3bc + 14ab^3c\), let's follow these steps:

1. **Identify the Greatest Common Factor (GCF)**:
   - The coefficients of the terms are 21 and 14. The GCF of 21 and 14 is 7.
   - For the variables:
     - \(a^3\) and \(a\) have a common factor of \(a\).
     - \(b\) and \(b^3\) have a common factor of \(b\).
     - \(c\) is common in both terms.

   So, the GCF for the entire expression is \(7abc\).

2. **Factor out the GCF**:
   - Divide each term by the GCF \(7abc\).

\[
21a^3bc + 14ab^3c = 7abc \left(\frac{21a^3bc}{7abc} + \frac{14ab^3c}{7abc}\right)
\]

\[
= 7abc \left(3a^2 + 2b^2\right)
\]

So, the factored form of \(21a^3bc + 14ab^3c\) is:

\[
\boxed{7abc(3a^2 + 2b^2)}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the greatest common factor (GCF) of a polynomial?
2. Can you factor the expression \(15x^2y + 25xy^2\)?
3. What is the difference between factoring and expanding expressions?
4. How can factoring be used to simplify algebraic fractions?
5. What are some special cases in factoring polynomials, like the difference of squares?

**Tip:** Always start by finding the GCF before attempting more complex factoring techniques.

Learn how to factor the polynomial expression 21a^3bc + 14ab^3c by identifying and factoring out the Greatest Common Factor (GCF). This solution breaks down each step, helping students in Grades 8-10 strengthen their algebraic factoring skills.

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