To solve the problem of finding the product of the polynomials [tex]\(6xy\)[/tex] and [tex]\(2x - y + 2\)[/tex], we perform polynomial multiplication step-by-step.
First, let's express the multiplication:
[tex]\[
(6xy) \cdot (2x - y + 2)
\][/tex]
We will distribute [tex]\(6xy\)[/tex] to each term of the polynomial [tex]\(2x - y + 2\)[/tex]:
[tex]\[
6xy \cdot (2x) + 6xy \cdot (-y) + 6xy \cdot (2)
\][/tex]
Let's perform each multiplication separately:
1. Multiplying [tex]\(6xy\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
6xy \cdot 2x = 12x^2y
\][/tex]
2. Multiplying [tex]\(6xy\)[/tex] by [tex]\(-y\)[/tex]:
[tex]\[
6xy \cdot (-y) = -6xy^2
\][/tex]
3. Multiplying [tex]\(6xy\)[/tex] by [tex]\(2\)[/tex]:
[tex]\[
6xy \cdot 2 = 12xy
\][/tex]
Now, we combine all the terms obtained from the distribution:
[tex]\[
12x^2y - 6xy^2 + 12xy
\][/tex]
Thus, the product of [tex]\(6xy\)[/tex] and [tex]\(2x - y + 2\)[/tex] is:
[tex]\[
12x^2y - 6xy^2 + 12xy
\][/tex]
Finally, we compare this result with the given options:
1. [tex]\(8x^2 - 4xy + 12x + y^2 - 2y\)[/tex]
2. [tex]\(12x^2 - 8xy + 12x + y^2 - 2y\)[/tex]
3. [tex]\(8x^2 + 4xy + 4x + y^2 - 2y\)[/tex]
4. [tex]\(12x^2 + 8xy + 4x + y^2 + 2y\)[/tex]
After inspecting the options, we notice that none of these match exactly with our multiplication result. Therefore, the correct option does not appear to be listed among the given choices.
If asked, the correct polynomial product should be:
[tex]\[
12x^2y - 6xy^2 + 12xy
\][/tex]
