Which shows one way to determine the factors of 4x3 + x2 – 8x – 2 by grouping?

Keep in mind that when factoring by grouping, we should divide the expression into two pairs that share some commonality. The goal is to create a common binomial factor after we factor each pair. It's important to mention that there can be multiple ways to pair the factors.

Let's explore the method of finding the factors of our expression through grouping:

Step 1. Identify shared multiples among the terms.

We can see that 4 and 8 are multiples of 2, 8 is a multiple of 4, and all are multiples of 1, suggesting various pairing options.

Step 2. Identify common variables.

Typically, we should pair variables that have the highest exponents to ensure that when factoring each group, we derive another common binomial factor.

Step 3. Factor each group; if it leads to another common binomial factor, you did it correctly. If it doesn't, try again.

Let's use these steps on our expression

For step 1, we are grouping and . Don't forget to pair the leftover factors too.

The common factor for the first group is , which we will factor out.

Look at that! We obtained the common binomial factor . Sadly, isn't among our provided choices, so we'll attempt this again:

This time, let's group the higher variables from step 2: and

The common factor in the first group turns out to be , and in the second one, it's -2, so we'll factor both of them out:

Now, we have chosen option A.

Therefore, we can conclude that the correct answer is: A) x2(4x + 1) – 2(4x + 1)