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The given function is:

\[
f(x) = 4x^2 + 16x
\]

To factor this completely, we follow these steps:

### Step 1: Identify the Greatest Common Factor (GCF)
The terms are \(4x^2\) and \(16x\), and the GCF between these two terms is \(4x\).

### Step 2: Factor out the GCF
We factor \(4x\) from both terms:

\[
f(x) = 4x(x + 4)
\]

So, the completely factored form of the function is:

\[
f(x) = 4x(x + 4)
\]

Would you like more details on the factorization process or have any other questions?

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Here are five related questions to expand your understanding:
1. How would you factor \(6x^2 + 18x\)?
2. What is the GCF of \(8x^3 + 12x^2\)?
3. How can you check if a factorization is correct?
4. What happens when you factor a quadratic function like \(x^2 + 5x + 6\)?
5. Can all quadratic functions be factored using real numbers?

**Tip**: Always start by factoring out the greatest common factor (GCF) before applying other factorization methods!

Factor the following completely: f(x) = 4x^2 + 16x

Learn how to factor the quadratic function 4x^2 + 16x by identifying the greatest common factor (GCF) and applying factoring techniques. This step-by-step solution is suitable for students in Grades 8-10.

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