To simplify the polynomial expression [tex]\((2x - 1)(2x^2 + 5x + 3) + (3x + 6)\)[/tex], let's break it down step by step, following the structure below:
1. Expand the first part of the expression:
[tex]\[
(2x - 1)(2x^2 + 5x + 3)
\][/tex]
2. Distribute each term in [tex]\((2x - 1)\)[/tex] across each term in [tex]\((2x^2 + 5x + 3)\)[/tex]:
[tex]\[
(2x - 1)(2x^2 + 5x + 3) = 2x \cdot (2x^2 + 5x + 3) - 1 \cdot (2x^2 + 5x + 3)
\][/tex]
3. Compute each part of the distribution separately:
- [tex]\(2x \cdot (2x^2) = 4x^3\)[/tex]
- [tex]\(2x \cdot (5x) = 10x^2\)[/tex]
- [tex]\(2x \cdot (3) = 6x\)[/tex]
- [tex]\(-1 \cdot (2x^2) = -2x^2\)[/tex]
- [tex]\(-1 \cdot (5x) = -5x\)[/tex]
- [tex]\(-1 \cdot (3) = -3\)[/tex]
4. Combine all the terms obtained:
[tex]\[
(2x - 1)(2x^2 + 5x + 3) = 4x^3 + 10x^2 + 6x - 2x^2 - 5x - 3
\][/tex]
5. Simplify by combining like terms (x^2 terms, x terms, constant terms):
[tex]\[
4x^3 + (10x^2 - 2x^2) + (6x - 5x) - 3
\][/tex]
[tex]\[
4x^3 + 8x^2 + x - 3
\][/tex]
6. Add the remaining part of the original expression [tex]\((3x + 6)\)[/tex] to this result:
[tex]\[
4x^3 + 8x^2 + x - 3 + 3x + 6
\][/tex]
7. Once again, combine like terms:
[tex]\[
4x^3 + 8x^2 + (x + 3x) + (-3 + 6)
\][/tex]
[tex]\[
4x^3 + 8x^2 + 4x + 3
\][/tex]
So, the fully simplified form of the polynomial [tex]\((2x - 1)(2x^2 + 5x + 3) + (3x + 6)\)[/tex] is:
[tex]\[
4x^3 + 8x^2 + 4x + 3
\][/tex]
