Let's break down the given expression step by step. We need to find the product of the two terms provided: [tex]\((-2, 3a^4b)\)[/tex] and [tex]\((1, 4b^2ca^3)\)[/tex].
Let's multiply the components of the pairs individually.
### Step 1: Multiply the numerical coefficients
First, we multiply the numerical coefficients of the two pairs. In this case, we multiply [tex]\(-2\)[/tex] and [tex]\(1\)[/tex]:
[tex]\[
-2 \times 1 = -2
\][/tex]
So, the numerical coefficient for the product will be [tex]\(-2\)[/tex].
### Step 2: Multiply the algebraic expressions
Now, we need to multiply the algebraic parts of the expressions: [tex]\(3a^4b\)[/tex] and [tex]\(4b^2ca^3\)[/tex].
To do this, we multiply the coefficients and then combine the powers of the same variables:
[tex]\[
3a^4b \times 4b^2ca^3 = (3 \times 4) \times (a^4 \times a^3) \times (b \times b^2) \times c
\][/tex]
Simplifying each part:
- Multiply the coefficients: [tex]\(3 \times 4 = 12\)[/tex]
- Combine the powers of [tex]\(a\)[/tex]: [tex]\(a^4 \times a^3 = a^{4+3} = a^7\)[/tex]
- Combine the powers of [tex]\(b\)[/tex]: [tex]\(b \times b^2 = b^{1+2} = b^3\)[/tex]
Putting it all together, we get:
[tex]\[
3a^4b \times 4b^2ca^3 = 12a^7b^3c
\][/tex]
### Step 3: Combine the results
Now, we combine the results from Step 1 and Step 2 to form our final answer:
[tex]\[
(-2, 3a^4b) \times (1, 4b^2ca^3) = (-2, 12a^7b^3c)
\][/tex]
So, the value of [tex]\(\epsilon\left(-2, 3a^4b\right) \left(1, 4b^2ca^3\right)\)[/tex] is:
[tex]\[
\boxed{(-2, 12a^7b^3c)}
\][/tex]
