Let's expand the expression [tex]\((-2x + 1)(x^2 - 9x + 10)\)[/tex] step by step.
### Step-by-Step Solution:
1. Distribute [tex]\(-2x\)[/tex] to each term in the polynomial [tex]\(x^2 - 9x + 10\)[/tex]:
[tex]\[
-2x \cdot (x^2 - 9x + 10) = -2x \cdot x^2 + (-2x) \cdot (-9x) + (-2x) \cdot 10
\][/tex]
Simplifying each term:
[tex]\[
-2x \cdot x^2 = -2x^3
\][/tex]
[tex]\[
-2x \cdot (-9x) = 18x^2
\][/tex]
[tex]\[
-2x \cdot 10 = -20x
\][/tex]
So, we get:
[tex]\[
-2x^3 + 18x^2 - 20x
\][/tex]
2. Distribute [tex]\(1\)[/tex] to each term in the polynomial [tex]\(x^2 - 9x + 10\)[/tex]:
[tex]\[
1 \cdot (x^2 - 9x + 10) = 1 \cdot x^2 + 1 \cdot (-9x) + 1 \cdot 10
\][/tex]
Simplifying each term:
[tex]\[
1 \cdot x^2 = x^2
\][/tex]
[tex]\[
1 \cdot (-9x) = -9x
\][/tex]
[tex]\[
1 \cdot 10 = 10
\][/tex]
So, we get:
[tex]\[
x^2 - 9x + 10
\][/tex]
3. Combine the results from both distributions:
[tex]\[
(-2x^3 + 18x^2 - 20x) + (x^2 - 9x + 10)
\][/tex]
4. Combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\(-2x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(18x^2 + x^2 = 19x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-20x - 9x = -29x\)[/tex]
- The constant term: [tex]\(10\)[/tex]
### Final Expanded Expression in Standard Form:
[tex]\[
-2x^3 + 19x^2 - 29x + 10
\][/tex]
So, the expanded expression is:
[tex]\[
-2x^3 + 19x^2 - 29x + 10
\][/tex]
