Sure, let's expand the expression [tex]\((x + 1)(3x^2 + 10x - 6)\)[/tex] step-by-step.
Step 1: Distribute [tex]\((x + 1)\)[/tex] to each term within the second polynomial [tex]\(3x^2 + 10x - 6\)[/tex].
This means breaking it down as follows:
[tex]\[
(x + 1)(3x^2 + 10x - 6) = x \cdot (3x^2 + 10x - 6) + 1 \cdot (3x^2 + 10x - 6)
\][/tex]
Step 2: Distribute [tex]\(x\)[/tex] across the terms inside the parentheses:
[tex]\[
x \cdot 3x^2 + x \cdot 10x + x \cdot (-6) = 3x^3 + 10x^2 - 6x
\][/tex]
Step 3: Distribute [tex]\(1\)[/tex] across the terms inside the parentheses:
[tex]\[
1 \cdot 3x^2 + 1 \cdot 10x + 1 \cdot (-6) = 3x^2 + 10x - 6
\][/tex]
Step 4: Combine the results from steps 2 and 3:
[tex]\[
3x^3 + 10x^2 - 6x + 3x^2 + 10x - 6
\][/tex]
Step 5: Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(10x^2 + 3x^2 = 13x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-6x + 10x = 4x\)[/tex]
So, putting it all together, we get:
[tex]\[
3x^3 + 13x^2 + 4x - 6
\][/tex]
Hence, the expanded form of the polynomial is:
[tex]\[
3x^3 + 13x^2 + 4x - 6
\][/tex]
