Certainly! Let's multiply the two given polynomials step by step:
The given problem is to multiply the polynomials [tex]\((2x + 5)\)[/tex] and [tex]\((3x^2 + 2x - 1)\)[/tex].
### Step-by-Step Solution:
1. Distribute [tex]\(2x\)[/tex] across each term in the second polynomial:
[tex]\[
2x \cdot (3x^2) + 2x \cdot (2x) + 2x \cdot (-1)
\][/tex]
- [tex]\(2x \cdot 3x^2 = 6x^3\)[/tex]
- [tex]\(2x \cdot 2x = 4x^2\)[/tex]
- [tex]\(2x \cdot (-1) = -2x\)[/tex]
So, the result of distributing [tex]\(2x\)[/tex] is:
[tex]\[
6x^3 + 4x^2 - 2x
\][/tex]
2. Distribute [tex]\(5\)[/tex] (the constant term) across each term in the second polynomial:
[tex]\[
5 \cdot (3x^2) + 5 \cdot (2x) + 5 \cdot (-1)
\][/tex]
- [tex]\(5 \cdot 3x^2 = 15x^2\)[/tex]
- [tex]\(5 \cdot 2x = 10x\)[/tex]
- [tex]\(5 \cdot (-1) = -5\)[/tex]
So, the result of distributing [tex]\(5\)[/tex] is:
[tex]\[
15x^2 + 10x - 5
\][/tex]
3. Combine all the terms obtained from both distributions:
[tex]\[
6x^3 + 4x^2 - 2x + 15x^2 + 10x - 5
\][/tex]
4. Combine like terms:
[tex]\[
\begin{aligned}
6x^3 & : \text{This term has no like term, so it remains } 6x^3. \\
4x^2 + 15x^2 & : Combine the \(x^2\) terms: \(4x^2 + 15x^2 = 19x^2\). \\
-2x + 10x & : Combine the \(x\) terms: \(-2x + 10x = 8x\). \\
-5 & : The constant term remains \(-5\).
\end{aligned}
\][/tex]
5. Write the final expression combining all the simplified terms:
[tex]\[
6x^3 + 19x^2 + 8x - 5
\][/tex]
Therefore, the final result of multiplying [tex]\((2x + 5)\)[/tex] with [tex]\((3x^2 + 2x - 1)\)[/tex] is:
[tex]\[
\boxed{6x^3 + 19x^2 + 8x - 5}
\][/tex]
