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Let's factor the given polynomial expression:

\[
-5x^3 - 5x^2 + 210x
\]

### Step 1: Factor out the greatest common factor (GCF).
The GCF of the terms \(-5x^3\), \(-5x^2\), and \(210x\) is \(-5x\), since each term contains \(x\) and \(-5\) divides all coefficients. Factoring out \(-5x\) from each term:

\[
-5x(x^2 + x - 42)
\]

### Step 2: Factor the quadratic expression \(x^2 + x - 42\).
We need to find two numbers that multiply to \(-42\) (the constant term) and add to \(1\) (the coefficient of \(x\)).

The pair of numbers is \(7\) and \(-6\), since \(7 \times (-6) = -42\) and \(7 + (-6) = 1\).

Thus, we can factor the quadratic:

\[
x^2 + x - 42 = (x + 7)(x - 6)
\]

### Step 3: Combine everything.
Now, include the factored GCF:

\[
-5x(x + 7)(x - 6)
\]

### Final factored form:
The completely factored form of the given expression is:

\[
\boxed{-5x(x + 7)(x - 6)}
\]

Would you like further clarification on any part of this solution?

Here are 5 related questions for practice:
1. How do you find the greatest common factor (GCF) of a polynomial?
2. What is the general process for factoring a quadratic expression?
3. How would you factor a cubic polynomial without a GCF?
4. How can you verify that a factored expression is correct?
5. What happens if the quadratic part is not factorable using integers?

**Tip:** Always start factoring by checking for a GCF. It simplifies the remaining steps!

Learn how to completely factor the polynomial expression -5x^3 - 5x^2 + 210x. This solution involves factoring out the greatest common factor (GCF) and breaking down a quadratic equation. Suitable for students in Grades 9-11.

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