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The given trinomial is:

\[
-3b^3 + 8b^2 + 3b
\]

### Step 1: Factor out the greatest common factor (GCF)
The GCF of the terms \(-3b^3\), \(8b^2\), and \(3b\) is \(b\). We can factor out \(b\) from the expression:

\[
b(-3b^2 + 8b + 3)
\]

### Step 2: Factor the quadratic expression
Now, we need to factor the quadratic expression inside the parentheses: \(-3b^2 + 8b + 3\).

We look for two numbers that multiply to give \((-3) \times 3 = -9\) and add to give 8. These numbers are 9 and -1.

Thus, we can rewrite the middle term:

\[
-3b^2 + 9b - b + 3
\]

### Step 3: Group and factor
Group the terms:

\[
(-3b^2 + 9b) - (b - 3)
\]

Factor each group:

\[
-3b(b - 3) - 1(b - 3)
\]

Now, factor out the common binomial factor \((b - 3)\):

\[
(b - 3)(-3b - 1)
\]

### Step 4: Final factored form
Thus, the complete factorization of the trinomial is:

\[
b(b - 3)(-3b - 1)
\]

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Would you like any further clarification or details on any part of the process? Here are some related questions to explore:

1. How do you identify the greatest common factor (GCF) in a polynomial?
2. What method can be used to factor a trinomial of degree 3 like this one?
3. Can you explain why grouping terms helps in factoring polynomials?
4. What are the benefits of factoring polynomials in algebraic expressions?
5. How can we verify the factorization of a trinomial?

**Tip:** Always check your factored form by multiplying it back to the original expression to ensure correctness.

Learn how to factor the trinomial -3b^3 + 8b^2 + 3b with a step-by-step explanation. This problem involves factoring out the greatest common factor (GCF) and factoring quadratic expressions. Suitable for students in Grades 8-10.

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