To factor the polynomial \( 2x^4 - 7x^3 + 6x^2 \), we can follow these steps:

### Step 1: Factor out the greatest common factor (GCF).
The GCF of the terms \( 2x^4 \), \( -7x^3 \), and \( 6x^2 \) is \( x^2 \). So, we can factor \( x^2 \) out:

\[
2x^4 - 7x^3 + 6x^2 = x^2(2x^2 - 7x + 6)
\]

### Step 2: Factor the quadratic expression.
Now we focus on factoring \( 2x^2 - 7x + 6 \). We need two numbers that multiply to \( 2 \times 6 = 12 \) and add up to \( -7 \). These numbers are \( -3 \) and \( -4 \).

So, we rewrite \( -7x \) as \( -3x - 4x \):

\[
2x^2 - 7x + 6 = 2x^2 - 3x - 4x + 6
\]

### Step 3: Group and factor by grouping.
Group the terms:

\[
(2x^2 - 3x) - (4x - 6)
\]

Now, factor each group:

\[
x(2x - 3) - 2(2x - 3)
\]

Since both groups contain \( (2x - 3) \), we can factor this out:

\[
(2x - 3)(x - 2)
\]

### Step 4: Write the final factored form.
Now that we have factored the quadratic, the entire expression becomes:

\[
2x^4 - 7x^3 + 6x^2 = x^2(2x - 3)(x - 2)
\]

Thus, the fully factored form is:

\[
x^2(2x - 3)(x - 2)
\]

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Do you want more details or have any questions about the factoring process?

### Related questions:
1. How do you factor a cubic polynomial?
2. What is the difference between factoring and expanding polynomials?
3. How do you find the greatest common factor (GCF) of polynomial terms?
4. How do you solve factored polynomials for real roots?
5. What are the applications of factoring polynomials in real life?

### Tip:  
When factoring, always start by checking for the greatest common factor (GCF), as it simplifies the process and prevents errors later on.

This page provides a step-by-step guide on how to factor the polynomial 2x⁴ - 7x³ + 6x². It covers concepts such as factoring by grouping, finding the greatest common factor (GCF), and factoring quadratic expressions. Ideal for students in Grades 9-12 who are learning advanced polynomial factoring techniques.

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