Factor completely.

4, x, cubed, minus, 28, x, squared, minus, 9, x, plus, 63

4*x*3−28*x*2−9*x*+63

Let's factor the expression:

\[
4x^3 - 28x^2 - 9x + 63
\]

### Step 1: Group terms
Group the terms to make factoring easier:

\[
(4x^3 - 28x^2) - (9x - 63)
\]

### Step 2: Factor out the greatest common factor (GCF) from each group
- From the first group \((4x^3 - 28x^2)\), the GCF is \(4x^2\), so:

\[
4x^2(x - 7)
\]

- From the second group \((-9x + 63)\), the GCF is \(-9\), so:

\[
-9(x - 7)
\]

### Step 3: Factor by grouping
Now we have:

\[
4x^2(x - 7) - 9(x - 7)
\]

Since \((x - 7)\) is common in both terms, we can factor it out:

\[
(x - 7)(4x^2 - 9)
\]

### Step 4: Factor the difference of squares
Notice that \(4x^2 - 9\) is a difference of squares:

\[
4x^2 - 9 = (2x - 3)(2x + 3)
\]

### Final factored form:
\[
(x - 7)(2x - 3)(2x + 3)
\]

This is the fully factored form of the expression.

Would you like further details or have any questions?

### Relative Questions:
1. How do you identify common factors when factoring?
2. What are the steps for factoring a difference of squares?
3. Why is grouping helpful in factoring polynomials?
4. Can all cubic polynomials be factored using grouping?
5. How can you verify that the factorization is correct?

### Tip: 
Always check for common factors before attempting more complex factoring methods to simplify the problem.

Factor completely: 4x^3 - 28x^2 - 9x + 63

Learn how to factor the cubic polynomial 4x^3 - 28x^2 - 9x + 63 with a detailed step-by-step guide. This solution covers polynomial factoring, grouping, and the difference of squares technique. Ideal for students in Grades 9-11 who are studying algebra.

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