To simplify the expression obtained by distributing the binomial [tex]\((2x + 3)\)[/tex] over the trinomial [tex]\((x^2 + x - 2)\)[/tex], we need to use the distributive property and combine like terms step by step. Let's start by expanding and then simplifying.
1. Distribute [tex]\(2x\)[/tex] across the trinomial [tex]\((x^2 + x - 2)\)[/tex]:
[tex]\[
(2x)(x^2) + (2x)(x) + (2x)(-2)
\][/tex]
- [tex]\((2x)(x^2) = 2x^3\)[/tex]
- [tex]\((2x)(x) = 2x^2\)[/tex]
- [tex]\((2x)(-2) = -4x\)[/tex]
So, the terms from distributing [tex]\(2x\)[/tex] are:
[tex]\[
2x^3 + 2x^2 - 4x
\][/tex]
2. Distribute [tex]\(3\)[/tex] across the trinomial [tex]\((x^2 + x - 2)\)[/tex]:
[tex]\[
(3)(x^2) + (3)(x) + (3)(-2)
\][/tex]
- [tex]\((3)(x^2) = 3x^2\)[/tex]
- [tex]\((3)(x) = 3x\)[/tex]
- [tex]\((3)(-2) = -6\)[/tex]
So, the terms from distributing [tex]\(3\)[/tex] are:
[tex]\[
3x^2 + 3x - 6
\][/tex]
3. Combine all the terms:
[tex]\[
2x^3 + 2x^2 - 4x + 3x^2 + 3x - 6
\][/tex]
4. Combine like terms:
- [tex]\(2x^3\)[/tex] (no like term for [tex]\(x^3\)[/tex])
- [tex]\(2x^2 + 3x^2 = 5x^2\)[/tex]
- [tex]\(-4x + 3x = -x\)[/tex]
- [tex]\(-6\)[/tex] (no like term for the constant)
So, the simplified expression is:
[tex]\[
2x^3 + 5x^2 - x - 6
\][/tex]
Therefore, the simplified product is:
[tex]\[
2x^3 + 5x^2 - x - 6
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{2 x^3 + 5 x^2 - x - 6}
\][/tex]
