Sure, let's simplify and evaluate the given polynomial step-by-step.
The given polynomial is:
[tex]\[ 2x^2 y + 3x y^2 - 3x y + 4x^2 y + y^2 - 5x y^2 \][/tex]
### Step 1: Combine Like Terms
First, let's combine the like terms:
- Combine the [tex]\( x^2 y \)[/tex] terms:
[tex]\[ 2x^2 y + 4x^2 y \][/tex]
[tex]\[ (2 + 4)x^2 y = 6x^2 y \][/tex]
- Combine the [tex]\( xy^2 \)[/tex] terms:
[tex]\[ 3x y^2 - 5x y^2 \][/tex]
[tex]\[ (3 - 5)x y^2 = -2x y^2 \][/tex]
- Combine the remaining terms:
- There is only one [tex]\( x y \)[/tex] term: [tex]\( -3x y \)[/tex]
- The [tex]\( y^2 \)[/tex] term remains as it is: [tex]\( y^2 \)[/tex]
So, the simplified polynomial is:
[tex]\[ 6x^2 y - 2x y^2 - 3x y + y^2 \][/tex]
### Step 2: Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -2 \)[/tex]
Now, we will substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -2 \)[/tex] into the simplified polynomial:
[tex]\[ 6(1)^2(-2) - 2(1)(-2)^2 - 3(1)(-2) + (-2)^2 \][/tex]
### Step 3: Calculate Each Term
Calculate the value of each term:
1. [tex]\( 6(1)^2(-2) \)[/tex]:
[tex]\[ 6 \cdot 1 \cdot -2 = -12 \][/tex]
2. [tex]\( -2(1)(-2)^2 \)[/tex]:
[tex]\[ -2 \cdot 1 \cdot 4 = -8 \][/tex]
3. [tex]\( -3(1)(-2) \)[/tex]:
[tex]\[ -3 \cdot 1 \cdot -2 = 6 \][/tex]
4. [tex]\( (-2)^2 \)[/tex]:
[tex]\[ 4 \][/tex]
### Step 4: Combine the Results
Add up all the computed values:
[tex]\[ -12 - 8 + 6 + 4 \][/tex]
[tex]\[ -12 - 8 = -20 \][/tex]
[tex]\[ -20 + 6 = -14 \][/tex]
[tex]\[ -14 + 4 = -10 \][/tex]
### Conclusion
The value of the polynomial [tex]\( 2 x^2 y + 3 x y^2 - 3 x y + 4 x^2 y + y^2 - 5 x y^2 \)[/tex] evaluated at [tex]\( x = 1 \)[/tex] and [tex]\( y = -2 \)[/tex] is:
[tex]\[ \boxed{-10} \][/tex]
