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------------------------------------------------ Factor completely:

[tex]\[ 4x^3 - 28x^2 + 40x \][/tex]

[tex]\[ 4x^3 - 28x^2 + 40x = \][/tex]

Sure! Let's factor the expression [tex]$4x^3 - 28x^2 + 40x$[/tex] completely, step by step.

### Step 1: Find the Greatest Common Factor (GCF)

First, we'll find the greatest common factor of the terms in the expression. The terms are [tex]$4x^3$[/tex], [tex]$-28x^2$[/tex], and [tex]$40x$[/tex].

1. The coefficients are 4, -28, and 40. The GCF of these numbers is 4.
2. Each term has at least one factor of [tex]$x$[/tex]. So, the GCF also includes [tex]$x$[/tex].

Thus, the GCF of the entire expression is [tex]$4x$[/tex].

### Step 2: Factor Out the GCF

Next, we'll factor out the GCF from each term:

[tex]\[
4x^3 - 28x^2 + 40x = 4x(x^2 - 7x + 10)
\][/tex]

### Step 3: Factor the Quadratic Expression

Now, we need to factor the quadratic expression [tex]$x^2 - 7x + 10$[/tex].

1. Look for two numbers that multiply to the constant term (10) and add up to the linear coefficient (-7). These numbers are -5 and -2.

2. So, we can write the quadratic as:
[tex]\[
x^2 - 7x + 10 = (x - 5)(x - 2)
\][/tex]

### Final Factored Form

Putting it all together, the fully factored form of the expression is:

[tex]\[
4x^3 - 28x^2 + 40x = 4x(x - 5)(x - 2)
\][/tex]

So, the completely factored expression is [tex]$4x(x - 5)(x - 2)$[/tex].