https://file.aiquickdraw.com/mathbot/file/1729567107071avd0vt2a.jpg

The given expression is:

\[
-2x^4 + 20x^3 + 48x^2
\]

### Step-by-Step Factorization:

#### Step 1: Factor out the **greatest common factor (GCF)**.
The GCF of \(-2x^4\), \(20x^3\), and \(48x^2\) is \(-2x^2\). Factoring out \(-2x^2\) gives:

\[
-2x^2 (x^2 - 10x - 24)
\]

#### Step 2: Factor the quadratic \(x^2 - 10x - 24\).
We need two numbers that multiply to \(-24\) (constant term) and add to \(-10\) (coefficient of \(x\)).

The pair of numbers is \(-12\) and \(2\). So:

\[
x^2 - 10x - 24 = (x - 12)(x + 2)
\]

#### Step 3: Write the fully factored expression.
Now substitute the factorization of the quadratic back into the expression:

\[
-2x^2 (x - 12)(x + 2)
\]

### Final Answer:
\[
-2x^2 (x - 12)(x + 2)
\]

Let me know if you need further clarification!

---

1. What is the role of the GCF in polynomial factorization?
2. How do you find two numbers that multiply to a constant and sum to a middle term?
3. What happens if a polynomial is not factorable by integers?
4. How can you confirm that your factorization is correct?
5. What if a polynomial involves higher-degree terms or more complex roots?

**Tip:** Always check for a common factor first before attempting more complicated factorization steps.

-2x^4 + 20x^3 + 48x^2. Factor completely.

Learn how to factor the polynomial -2x^4 + 20x^3 + 48x^2 completely. This problem involves using the greatest common factor (GCF) and quadratic factoring, providing a detailed solution for high school algebra students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation