Answer :
Sure! Let's multiply the two expressions [tex]\((x^2 - 5x)\)[/tex] and [tex]\((2x^2 + x - 3)\)[/tex] step by step to find the expanded form.
### Step-by-Step Expansion:
1. Distribute [tex]\(x^2\)[/tex] to each term in the second expression:
- [tex]\(x^2 \times 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \times x = x^3\)[/tex]
- [tex]\(x^2 \times (-3) = -3x^2\)[/tex]
Combining these, we get:
[tex]\[
2x^4 + x^3 - 3x^2
\][/tex]
2. Distribute [tex]\(-5x\)[/tex] to each term in the second expression:
- [tex]\(-5x \times 2x^2 = -10x^3\)[/tex]
- [tex]\(-5x \times x = -5x^2\)[/tex]
- [tex]\(-5x \times (-3) = 15x\)[/tex]
Combining these, we get:
[tex]\[
-10x^3 - 5x^2 + 15x
\][/tex]
3. Combine like terms from both distributions:
- For [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(15x\)[/tex]
So, the complete expanded expression is:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
The correct option that matches this expanded expression is A. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
I hope this helps you understand how to expand and simplify polynomial expressions! If you have any more questions, feel free to ask.
### Step-by-Step Expansion:
1. Distribute [tex]\(x^2\)[/tex] to each term in the second expression:
- [tex]\(x^2 \times 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \times x = x^3\)[/tex]
- [tex]\(x^2 \times (-3) = -3x^2\)[/tex]
Combining these, we get:
[tex]\[
2x^4 + x^3 - 3x^2
\][/tex]
2. Distribute [tex]\(-5x\)[/tex] to each term in the second expression:
- [tex]\(-5x \times 2x^2 = -10x^3\)[/tex]
- [tex]\(-5x \times x = -5x^2\)[/tex]
- [tex]\(-5x \times (-3) = 15x\)[/tex]
Combining these, we get:
[tex]\[
-10x^3 - 5x^2 + 15x
\][/tex]
3. Combine like terms from both distributions:
- For [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(15x\)[/tex]
So, the complete expanded expression is:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
The correct option that matches this expanded expression is A. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
I hope this helps you understand how to expand and simplify polynomial expressions! If you have any more questions, feel free to ask.
