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The given expression is:

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

### Step 1: Factor out the greatest common factor (GCF)
The GCF of the terms \(-5x^3\), \(-5x^2\), and \(210x\) is \(-5x\). Let's factor that out:

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

### Step 2: Factor the quadratic expression \(x^2 + x - 42\)
We now need to factor \(x^2 + x - 42\). To factor it, we need two numbers that multiply to \(-42\) and add to \(1\) (the coefficient of \(x\)).

These numbers are \(7\) and \(-6\), because:

\[
7 \times (-6) = -42 \quad \text{and} \quad 7 + (-6) = 1
\]

Thus, we can factor the quadratic expression as:

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

### Step 3: Final factored form
Now, substituting this back, the completely factored form of the expression is:

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

This is the fully factored form of the given expression.

Would you like more details or explanations on any steps? Here are 5 related questions you might consider:

1. How do you find the greatest common factor (GCF) of multiple terms?
2. What is the process for factoring quadratic expressions?
3. How can you check if a factorization is correct?
4. What are special cases in factoring polynomials?
5. How does factoring help in solving equations?

**Tip:** Always double-check by multiplying the factors to ensure the original expression is recovered.

Learn how to factor the polynomial expression -5x^3 - 5x^2 + 210x completely. This solution covers factoring out the greatest common factor (GCF) and factoring quadratic expressions. Ideal for students in grades 9-12 studying algebra.

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