Subtract the sum of `(x^2+2y^2+7x^2y) and (x^2-3xy^2)` from the sum of `(x^2-y^2-xy) and (3x^2-2y^2-7)`

Answer

To solve the problem, we will follow the steps outlined in the video transcript. We need to perform the necessary additions and subtractions of the given algebraic expressions.

### Step-by-step Solution:

1. **Identify the expressions:**
- First expression (E1): \( x^2 + 2y^2 + 7x^2y \)
- Second expression (E2): \( x^2 - 3xy^2 \)
- Third expression (E3): \( x^2 - y^2 - xy \)
- Fourth expression (E4): \( 3x^2 - 2y^2 - 7 \)

2. **Calculate the sum of E3 and E4:**
\[ E3 + E4 = (x^2 - y^2 - xy) + (3x^2 - 2y^2 - 7) \]
Combine like terms:
\[ = (x^2 + 3x^2) + (-y^2 - 2y^2) + (-xy) + (-7) \]
\[ = 4x^2 - 3y^2 - xy - 7 \]

3. **Calculate the sum of E1 and E2:**
\[ E1 + E2 = (x^2 + 2y^2 + 7x^2y) + (x^2 - 3xy^2) \]
Combine like terms:
\[ = (x^2 + x^2) + (2y^2) + (7x^2y) + (-3xy^2) \]
\[ = 2x^2 + 2y^2 + 7x^2y - 3xy^2 \]

4. **Subtract the sum of E1 and E2 from the sum of E3 and E4:**
\[ (E3 + E4) - (E1 + E2) = (4x^2 - 3y^2 - xy - 7) - (2x^2 + 2y^2 + 7x^2y - 3xy^2) \]
Distributing the negative sign:
\[ = 4x^2 - 3y^2 - xy - 7 - 2x^2 - 2y^2 - 7x^2y + 3xy^2 \]
Combine like terms:
\[ = (4x^2 - 2x^2) + (-3y^2 - 2y^2) + (-xy) + (-7) + (-7x^2y) + (3xy^2) \]
\[ = 2x^2 - 5y^2 - xy - 7 - 7x^2y + 3xy^2 \]

5. **Final expression:**
\[ = 2x^2 - 5y^2 - xy - 7 - 7x^2y + 3xy^2 \]

### Final Answer:
\[ 2x^2 - 5y^2 - xy - 7 - 7x^2y + 3xy^2 \]