To analyze the polynomial expression \(2x^3 + 3x^2 - 5x\), we can perform several operations such as factoring, finding its roots, and determining its critical points. Let's go through these steps one by one.

### 1. Factoring the Polynomial

First, we can factor out the greatest common factor (GCF) from the terms:

\[ 2x^3 + 3x^2 - 5x = x(2x^2 + 3x - 5) \]

Now, we need to factor the quadratic expression \(2x^2 + 3x - 5\). To do this, we look for two numbers that multiply to \(2 \cdot (-5) = -10\) and add up to \(3\). These numbers are \(5\) and \(-2\):

\[ 2x^2 + 3x - 5 = 2x^2 + 5x - 2x - 5 \]

Next, we group the terms and factor by grouping:

\[ 2x^2 + 5x - 2x - 5 = x(2x + 5) - 1(2x + 5) \]

\[ = (x - 1)(2x + 5) \]

So, the fully factored form of the polynomial is:

\[ 2x^3 + 3x^2 - 5x = x(x - 1)(2x + 5) \]

### 2. Finding the Roots

To find the roots of the polynomial, we set the factored form equal to zero and solve for \(x\):

\[ x(x - 1)(2x + 5) = 0 \]

This gives us three roots:

\[ x = 0 \]
\[ x - 1 = 0 \implies x = 1 \]
\[ 2x + 5 = 0 \implies x = -\frac{5}{2} \]

So, the roots of the polynomial are \(x = 0\), \(x = 1\), and \(x = -\frac{5}{2}\).

### 3. Determining Critical Points

To find the critical points, we need to take the derivative of the polynomial and set it equal to zero:

\[ f(x) = 2x^3 + 3x^2 - 5x \]

\[ f'(x) = \frac{d}{dx}(2x^3 + 3x^2 - 5x) \]

\[ f'(x) = 6x^2 + 6x - 5 \]

Set the derivative equal to zero to find the critical points:

\[ 6x^2 + 6x - 5 = 0 \]

We can solve this quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 6\), \(b = 6\), and \(c = -5\):

\[ x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 6 \cdot (-5)}}{2 \cdot 6} \]

\[ x = \frac{-6 \pm \sqrt{36 + 120}}{12} \]

\[ x = \frac{-6 \pm \sqrt{156}}{12} \]

\[ x = \frac{-6 \pm 2\sqrt{39}}{12} \]

\[ x = \frac{-3 \pm \sqrt{39}}{6} \]

So, the critical points are:

\[ x = \frac{-3 + \sqrt{39}}{6} \]
\[ x = \frac{-3 - \sqrt{39}}{6} \]

These are the critical points of the polynomial.

### Summary

- The factored form of the polynomial is \(x(x - 1)(2x + 5)\).
- The roots of the polynomial are \(x = 0\), \(x = 1\), and \(x = -\frac{5}{2}\).
- The critical points are \(x = \frac{-3 + \sqrt{39}}{6}\) and \(x = \frac{-3 - \sqrt{39}}{6}\).