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We are asked to factor the polynomial \( P(x) = -5x^3 - 4x^2 + x \) and find its zeros.

### Step 1: Factor the polynomial.
The polynomial has a common factor of \( x \) in all its terms. Factoring \( x \) out:

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

### Step 2: Factor the quadratic expression.
Now, we need to factor \( -5x^2 - 4x + 1 \). To factor this, we look for two numbers that multiply to \( (-5)(1) = -5 \) and add to \( -4 \). The two numbers are \( -5 \) and \( 1 \).

So, we rewrite the middle term \( -4x \) as \( -5x + x \):

\[
-5x^2 - 4x + 1 = -5x^2 - 5x + x + 1
\]

Next, we factor by grouping:

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

### Step 3: Write the fully factored form.
Now, the fully factored form of the polynomial is:

\[
P(x) = x(x + 1)(-5x + 1)
\]

### Step 4: Find the zeros.
To find the zeros, set each factor equal to zero:

1. \( x = 0 \)
2. \( x + 1 = 0 \) ⟶ \( x = -1 \)
3. \( -5x + 1 = 0 \) ⟶ \( x = \frac{1}{5} \)

### Conclusion:
The zeros of the polynomial are:

\[
x = 0, \quad x = -1, \quad x = \frac{1}{5}
\]

Would you like more details or explanations on any of these steps?

Here are five related questions:
1. How would you factor a polynomial that doesn't have a common factor?
2. What is the importance of factoring in finding the zeros of a polynomial?
3. Can all cubic polynomials be factored easily, or are some irreducible?
4. How do you handle higher-degree polynomials when factoring?
5. How do the zeros of a polynomial relate to its graph?

**Tip:** Always check for a common factor first when factoring any polynomial—it simplifies the entire process.

Factor the polynomial and use the factored form to find the zeros: P(x) = -5x^3 - 4x^2 + x

Learn how to factor the cubic polynomial -5x^3 - 4x^2 + x and find its zeros with a detailed step-by-step solution. This problem explores key algebraic concepts, including factoring by grouping and the zero-product property. Suitable for students in Grades 10-12.

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