To simplify the given expression [tex]\(\left(9 x^4 y^3 + 10 x^2 y + 2 x y\right) - \left(7 x^4 y^3 - 5 x^2 y + 7 x y\right)\)[/tex], we will follow these steps:
1. Distribute the negative sign to every term in the second expression.
2. Combine like terms by finding similar terms in the resulting expression and performing the indicated operations.
3. Factor the expression if possible.
Let's break it down step-by-step:
### Step 1: Distribute the negative sign
First, distribute the negative sign across each term in the second group:
[tex]\[
(9 x^4 y^3 + 10 x^2 y + 2 x y) - (7 x^4 y^3 - 5 x^2 y + 7 x y)
\][/tex]
Becomes:
[tex]\[
9 x^4 y^3 + 10 x^2 y + 2 x y - 7 x^4 y^3 + 5 x^2 y - 7 x y
\][/tex]
### Step 2: Combine like terms
Now, we combine the like terms:
#### For [tex]\(x^4 y^3\)[/tex]:
[tex]\[
9 x^4 y^3 - 7 x^4 y^3 = (9 - 7) x^4 y^3 = 2 x^4 y^3
\][/tex]
#### For [tex]\(x^2 y\)[/tex]:
[tex]\[
10 x^2 y + 5 x^2 y = (10 + 5) x^2 y = 15 x^2 y
\][/tex]
#### For [tex]\(x y\)[/tex]:
[tex]\[
2 x y - 7 x y = (2 - 7) x y = -5 x y
\][/tex]
Putting these combined terms together, we get:
[tex]\[
2 x^4 y^3 + 15 x^2 y - 5 x y
\][/tex]
### Step 3: Factor the expression
We notice that each term has a common factor of [tex]\(x y\)[/tex], so we can factor out [tex]\(x y\)[/tex]:
[tex]\[
x y (2 x^3 y^2 + 15 x - 5)
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{x y (2 x^3 y^2 + 15 x - 5)}
\][/tex]
This is our final answer in the order of descending exponents of [tex]\(x\)[/tex].
