The given expression is \( 2x^3 - 4x^2 + 6x \). Since it is a polynomial, we can analyze it in various ways, such as simplifying, factoring, or finding its derivative or integral. Below are some possibilities:

### 1. **Factoring the Expression**:
We can factor out the greatest common factor (GCF) from all the terms:

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

This is the fully factored form. The quadratic part \( x^2 - 2x + 3 \) does not factor further using real numbers.

### 2. **Finding the Derivative**:
If you want to find the derivative of the expression with respect to \(x\):

\[
f(x) = 2x^3 - 4x^2 + 6x
\]
The derivative is:
\[
f'(x) = 6x^2 - 8x + 6
\]

### 3. **Finding the Integral**:
If you want to find the indefinite integral of the expression:

\[
\int (2x^3 - 4x^2 + 6x) \, dx = \frac{2x^4}{4} - \frac{4x^3}{3} + \frac{6x^2}{2} + C
\]

Simplifying:
\[
\int (2x^3 - 4x^2 + 6x) \, dx = \frac{x^4}{2} - \frac{4x^3}{3} + 3x^2 + C
\]
Where \( C \) is the constant of integration.

Would you like more details or have any specific questions?

### 5 Related Questions:
1. How do we find the roots of the polynomial \( 2x^3 - 4x^2 + 6x \)?
2. Can we solve \( 2x^3 - 4x^2 + 6x = 0 \) for \( x \)?
3. What are the critical points of the function \( f(x) = 2x^3 - 4x^2 + 6x \)?
4. How can we graph \( f(x) = 2x^3 - 4x^2 + 6x \)?
5. What is the concavity of \( f(x) = 2x^3 - 4x^2 + 6x \)?

### Tip:
When factoring, always start by factoring out the greatest common factor (GCF) before checking if further factorization is possible.

Explore the polynomial 2x^3 - 4x^2 + 6x with solutions including factoring, finding its derivative, and solving its integral. This problem covers key calculus concepts and is suitable for students in Grades 11-12.

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