Certainly! Let's go through the multiplication step-by-step.
Given the expressions to multiply:
[tex]\[ (2x - 1) \left( 3x^2 + x - 2 \right) \][/tex]
We'll multiply each term in the first expression by each term in the second expression and then combine like terms.
### Step 1: Distribute each term in the first expression to each term in the second expression
Let's start by distributing [tex]\(2x\)[/tex]:
[tex]\[ 2x \cdot 3x^2 = 6x^3 \][/tex]
[tex]\[ 2x \cdot x = 2x^2 \][/tex]
[tex]\[ 2x \cdot (-2) = -4x \][/tex]
Now distribute [tex]\(-1\)[/tex]:
[tex]\[ (-1) \cdot 3x^2 = -3x^2 \][/tex]
[tex]\[ (-1) \cdot x = -x \][/tex]
[tex]\[ (-1) \cdot (-2) = 2 \][/tex]
### Step 2: Combine all these distributed terms
After distributing, we'll have the following terms:
[tex]\[ 6x^3, 2x^2, -4x, -3x^2, -x, 2 \][/tex]
### Step 3: Combine like terms
Combine the [tex]\(x^2\)[/tex] terms and the [tex]\(x\)[/tex] terms:
[tex]\[ 6x^3 + 2x^2 - 3x^2 - 4x - x + 2 \][/tex]
Simplify by summing the coefficients of like terms:
[tex]\[ (6x^3) + (2x^2 - 3x^2) + (-4x - x) + 2 = 6x^3 - x^2 - 5x + 2 \][/tex]
### Final Answer
So the product of [tex]\((2x - 1) \left( 3x^2 + x - 2 \right) \)[/tex] is:
[tex]\[ 6x^3 - x^2 - 5x + 2 \][/tex]
