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To factor the given polynomial \( 8x^3 + 4x^2 - 10x - 5 \) by grouping, we will break it into groups and factor common terms. Here's the step-by-step process:

### Step 1: Group the terms
Group the polynomial into two pairs:

\[
(8x^3 + 4x^2) + (-10x - 5)
\]

### Step 2: Factor out the greatest common factor (GCF) from each group

- From the first group \( 8x^3 + 4x^2 \), the GCF is \( 4x^2 \), so factor that out:

\[
4x^2(2x + 1)
\]

- From the second group \( -10x - 5 \), the GCF is \( -5 \), so factor that out:

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

### Step 3: Combine the factored terms

Now, notice that both terms contain \( (2x + 1) \). Factor that out:

\[
(2x + 1)(4x^2 - 5)
\]

### Final factored form:

\[
(2x + 1)(4x^2 - 5)
\]

This is the fully factored form of the given polynomial.

Would you like further details or explanations on any step?

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Here are some related questions to explore further:

1. How can you factor a polynomial with more than four terms by grouping?
2. What if the terms do not initially factor neatly—how do you handle factoring by grouping in such cases?
3. What is the difference between factoring by grouping and factoring using the distributive property?
4. Can \(4x^2 - 5\) be factored further in some cases? (Hint: look for special factorization forms.)
5. How do you verify that your factorization is correct?

**Tip:** Always check for the greatest common factor (GCF) first before attempting more complex factoring methods.

Factor the polynomial 8x^3 + 4x^2 - 10x - 5 by grouping.

Learn how to factor the polynomial 8x^3 + 4x^2 - 10x - 5 by grouping. This solution explains each step, including identifying the greatest common factor and using the distributive property to simplify the polynomial. Perfect for high school algebra students.

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