To expand the expression $( -x + 2)(x^2 + 9x - 2)$ into a polynomial in standard form, we can use the distributive property by multiplying each term in $(-x + 2)$ with each term in $(x^2 + 9x - 2)$.

Step-by-Step Solution

1. Distribute $-x$ across $(x^2 + 9x - 2)$: $-x \cdot x^2 = -x^3$ $-x \cdot 9x = -9x^2$ $-x \cdot (-2) = 2x$

2. Distribute $2$ across $(x^2 + 9x - 2)$: $2 \cdot x^2 = 2x^2$ $2 \cdot 9x = 18x$ $2 \cdot (-2) = -4$

3. Combine all terms: $-x^3 - 9x^2 + 2x + 2x^2 + 18x - 4$

4. Combine like terms:

   • The $x^3$ term: $-x^3$
   • The $x^2$ terms: $-9x^2 + 2x^2 = -7x^2$
   • The $x$ terms: $2x + 18x = 20x$
   • The constant term: $-4$

5. Write the polynomial in standard form: $-x^3 - 7x^2 + 20x - 4$

Final Answer:

$- x^3 - 7x^2 + 20x - 4$

Would you like further details or have any questions?

Here are some related questions to explore further:

1. How can the distributive property be applied in different polynomial multiplications?
2. What techniques simplify expanding binomials and trinomials?
3. How does combining like terms help in simplifying polynomial expressions?
4. How do you find the degree of a polynomial after expansion?
5. How does this process change when expanding higher-degree polynomials?

Tip: Always arrange terms in descending order of powers for a polynomial in standard form.