To solve the given expression [tex]\((2x^2 y^3 + 12x y^2) \cdot (-12xy + 4y^2 - 14x^3 y)\)[/tex], we need to expand and simplify it. Let's proceed step by step.
Step 1: Distribute each term of the first polynomial by each term of the second polynomial.
Given the expression:
[tex]\[
(2x^2 y^3 + 12x y^2) \cdot (-12xy + 4y^2 - 14x^3 y)
\][/tex]
We need to distribute the two terms from the first polynomial across each of the three terms in the second polynomial.
Let's start with [tex]\(2x^2 y^3 \)[/tex]:
[tex]\[
2x^2 y^3 \cdot (-12xy) + 2x^2 y^3 \cdot 4y^2 + 2x^2 y^3 \cdot (-14x^3 y)
\][/tex]
Now separately:
1. [tex]\( 2x^2 y^3 \cdot (-12xy) \)[/tex]:
[tex]\[
2x^2 y^3 \cdot -12xy = 2 \cdot (-12) \cdot x^2 \cdot x \cdot y^3 \cdot y = -24x^3 y^4
\][/tex]
2. [tex]\( 2x^2 y^3 \cdot 4y^2 \)[/tex]:
[tex]\[
2x^2 y^3 \cdot 4y^2 = 2 \cdot 4 \cdot x^2 \cdot y^3 \cdot y^2 = 8x^2 y^5
\][/tex]
3. [tex]\( 2x^2 y^3 \cdot (-14x^3 y) \)[/tex]:
[tex]\[
2x^2 y^3 \cdot -14x^3 y = 2 \cdot (-14) \cdot x^2 \cdot x^3 \cdot y^3 \cdot y = -28x^5 y^4
\][/tex]
Now, let's proceed with [tex]\(12xy^2\)[/tex]:
[tex]\[
12x y^2 \cdot (-12xy) + 12x y^2 \cdot 4y^2 + 12x y^2 \cdot (-14x^3 y)
\][/tex]
Separately:
1. [tex]\( 12x y^2 \cdot (-12xy) \)[/tex]:
[tex]\[
12x y^2 \cdot -12xy = 12 \cdot (-12) \cdot x \cdot x \cdot y^2 \cdot y = -144x^2 y^3
\][/tex]
2. [tex]\( 12x y^2 \cdot 4y^2 \)[/tex]:
[tex]\[
12x y^2 \cdot 4y^2 = 12 \cdot 4 \cdot x \cdot y^2 \cdot y^2 = 48x y^4
\][/tex]
3. [tex]\( 12x y^2 \cdot (-14x^3 y) \)[/tex]:
[tex]\[
12x y^2 \cdot -14x^3 y = 12 \cdot (-14) \cdot x \cdot x^3 \cdot y^2 \cdot y = -168x^4 y^3
\][/tex]
Step 2: Combine all the terms.
Now, we combine all the resulting terms:
[tex]\[
-24x^3 y^4 + 8x^2 y^5 - 28x^5 y^4 - 144x^2 y^3 + 48x y^4 - 168x^4 y^3
\][/tex]
We rearrange and sort these terms:
[tex]\[
-28x^5 y^4 - 168x^4 y^3 - 24x^3 y^4 + 8x^2 y^5 - 144x^2 y^3 + 48x y^4
\][/tex]
This is the expanded and simplified form of the given expression.
