Directions - Multiply the binomials using the box method.

To multiply the binomials \((3x^3 + 2x)(x^3 - 5)\) using the box method, follow these steps:

### Step 1: Set Up the Box
We will create a 2x2 box since both expressions are binomials.

```
| x^3 | -5
---------|----------|-------
3x^3 | |
---------|----------|-------
2x | |
```

### Step 2: Fill in the Box
Now we will multiply each term in the first binomial \((3x^3 + 2x)\) with each term in the second binomial \((x^3 - 5)\):

1. Top left: \( 3x^3 \cdot x^3 = 3x^6 \)
2. Top right: \( 3x^3 \cdot -5 = -15x^3 \)
3. Bottom left: \( 2x \cdot x^3 = 2x^4 \)
4. Bottom right: \( 2x \cdot -5 = -10x \)

Now our box is filled:

```
| x^3 | -5
---------|----------|-------
3x^3 | 3x^6 | -15x^3
---------|----------|-------
2x | 2x^4 | -10x
```

### Step 3: Combine Like Terms
Now we write down all the terms inside the box:

- \(3x^6\)
- \(-15x^3\)
- \(2x^4\)
- \(-10x\)

Now we combine them:

The final result is:
\[
3x^6 + 2x^4 - 15x^3 - 10x
\]

### Conclusion
So, the product of \((3x^3 + 2x)(x^3 - 5)\) is:
\[
\boxed{3x^6 + 2x^4 - 15x^3 - 10x}
\]