Alright, let’s multiply the two polynomials [tex]\((4x^2 - 5x + 3)\)[/tex] and [tex]\((5x^2 + 3x + 2)\)[/tex] step-by-step.
Step 1: Distribute each term of the first polynomial to every term of the second polynomial.
Here’s how you can do it:
1. Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\(5x^2 + 3x + 2\)[/tex]:
[tex]\[
4x^2 \cdot 5x^2 = 20x^4
\][/tex]
[tex]\[
4x^2 \cdot 3x = 12x^3
\][/tex]
[tex]\[
4x^2 \cdot 2 = 8x^2
\][/tex]
2. Multiply [tex]\(-5x\)[/tex] by each term in [tex]\(5x^2 + 3x + 2\)[/tex]:
[tex]\[
-5x \cdot 5x^2 = -25x^3
\][/tex]
[tex]\[
-5x \cdot 3x = -15x^2
\][/tex]
[tex]\[
-5x \cdot 2 = -10
\][/tex]
3. Multiply [tex]\(3\)[/tex] by each term in [tex]\(5x^2 + 3x + 2\)[/tex]:
[tex]\[
3 \cdot 5x^2 = 15x^2
\][/tex]
[tex]\[
3 \cdot 3x = 9x
\][/tex]
[tex]\[
3 \cdot 2 = 6
\][/tex]
Step 2: Sum up all the terms.
Now, write down all the obtained terms and group them by the same powers of [tex]\(x\)[/tex]:
[tex]\[
20x^4 + 12x^3 + 8x^2 - 25x^3 - 15x^2 -10x + 15x^2 + 9x + 6
\][/tex]
Combine the like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(20x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(12x^3 - 25x^3 = -13x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(8x^2 - 15x^2 + 15x^2 = 8x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-10x + 9x = -x\)[/tex]
- Constants: [tex]\(6\)[/tex]
So, the resulting polynomial is:
[tex]\[
20x^4 - 13x^3 + 8x^2 - x + 6
\][/tex]
This is the final answer:
[tex]\[
20x^4 - 13x^3 + 8x^2 - x + 6
\][/tex]
