factoring solution for given problem. 3x³+2x²-5x

To factor the expression \(3x^3 + 2x^2 - 5x\), follow these steps:

1. **Identify the greatest common factor (GCF):**
   The GCF of the terms \(3x^3\), \(2x^2\), and \(-5x\) is \(x\). Factor out \(x\) from the entire expression:
   \[
   3x^3 + 2x^2 - 5x = x(3x^2 + 2x - 5)
   \]

2. **Factor the quadratic trinomial:**
   Next, we need to factor the quadratic expression \(3x^2 + 2x - 5\). To do this, we look for two numbers that multiply to \(3 \times (-5) = -15\) and add to \(2\).

   These numbers are \(5\) and \(-3\), because:
   \[
   5 \times (-3) = -15 \quad \text{and} \quad 5 + (-3) = 2
   \]

3. **Rewrite the middle term:**
   Rewrite the middle term \(2x\) using the numbers found:
   \[
   3x^2 + 5x - 3x - 5
   \]

4. **Factor by grouping:**
   Group the terms and factor each group:
   \[
   (3x^2 + 5x) - (3x + 5)
   \]
   Factor out the common factors from each group:
   \[
   x(3x + 5) - 1(3x + 5)
   \]

5. **Factor out the common binomial factor:**
   Now, factor out the common binomial \(3x + 5\):
   \[
   (x - 1)(3x + 5)
   \]

6. **Include the GCF:**
   Don't forget the GCF \(x\) that we factored out initially:
   \[
   3x^3 + 2x^2 - 5x = x(x - 1)(3x + 5)
   \]

Thus, the fully factored form of the expression is:
\[
\boxed{x(x - 1)(3x + 5)}
\]

Would you like more details on any step? Here are some related questions to further explore:

1. What are the steps to factor a cubic expression?
2. How can you check the factors of a quadratic trinomial?
3. What is the significance of the greatest common factor (GCF) in factoring?
4. Are there special cases in factoring where different methods are required?
5. How would you factor a cubic expression with more complex coefficients?

**Tip:** When factoring, always start by factoring out the GCF, as it simplifies the problem and makes further factoring easier.

Learn how to factor the polynomial 3x³ + 2x² - 5x with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomial factoring, quadratic equations, and factoring by grouping. Suitable for students in Grades 10-12.

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