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To factor the polynomial \( 14x^2 + 28x^5 \) by its greatest common monomial factor, we need to identify the greatest common factor (GCF) of the coefficients and the variable parts.

### Step-by-Step Solution:
1. **Identify the GCF of the coefficients**: 
   - The coefficients are 14 and 28.
   - The GCF of 14 and 28 is 14.

2. **Identify the GCF of the variable parts**:
   - The variable parts are \( x^2 \) and \( x^5 \).
   - The GCF is the smallest power of \( x \), which is \( x^2 \).

3. **Combine the GCF of the coefficients and the variable parts**:
   - The GCF of the entire polynomial is \( 14x^2 \).

4. **Factor out the GCF**:
   - Factoring \( 14x^2 \) out of \( 14x^2 + 28x^5 \):
   \[
   14x^2 + 28x^5 = 14x^2(1 + 2x^3)
   \]

So, the factored form of the polynomial is:

\[
14x^2(1 + 2x^3)
\]

Would you like more details on any specific part of the solution? Here are some related questions:

1. What is the process to find the GCF of two polynomials?
2. How do you factor polynomials with more than two terms?
3. What are the steps to simplify expressions after factoring out the GCF?
4. How can factoring be applied in solving polynomial equations?
5. What are some common errors to watch out for when factoring polynomials?

**Tip:** Always double-check your factoring by expanding the factored form to ensure it matches the original polynomial.

Factor the polynomial 14x^2 + 28x^5 by its greatest common monomial factor.

Learn how to factor the polynomial 14x^2 + 28x^5 by finding the greatest common factor (GCF). This step-by-step solution breaks down the process of identifying the GCF of both coefficients and variable parts, suitable for students in Grades 8-10.

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