What must be subtracted from 4x²y²+3xy+3xy² to obtain 8xy²-4x²y+7x²y²+10xy

We are tasked with finding the expression that must be subtracted from \( 4x^2y^2 + 3xy + 3xy^2 \) to obtain \( 8xy^2 - 4x^2y + 7x^2y^2 + 10xy \).

Let the expression we need to subtract be \( E \). Therefore, we can write the equation as:

\[
(4x^2y^2 + 3xy + 3xy^2) - E = 8xy^2 - 4x^2y + 7x^2y^2 + 10xy
\]

Now, solve for \( E \):

\[
E = (4x^2y^2 + 3xy + 3xy^2) - (8xy^2 - 4x^2y + 7x^2y^2 + 10xy)
\]

Distribute the negative sign on the right-hand side:

\[
E = 4x^2y^2 + 3xy + 3xy^2 - 8xy^2 + 4x^2y - 7x^2y^2 - 10xy
\]

Now, group like terms:

\[
E = (4x^2y^2 - 7x^2y^2) + (3xy - 10xy) + (3xy^2 - 8xy^2) + 4x^2y
\]

Simplify each group:

\[
E = -3x^2y^2 - 7xy - 5xy^2 + 4x^2y
\]

Therefore, the expression that must be subtracted is:

\[
E = -3x^2y^2 + 4x^2y - 5xy^2 - 7xy
\]

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you approach solving equations involving multiple variables?
2. What are some common mistakes when distributing negative signs?
3. How can you verify the result by substituting specific values for \( x \) and \( y \)?
4. What is the significance of grouping like terms in algebraic expressions?
5. How do you handle subtraction of polynomials in general?

**Tip**: Always carefully handle negative signs when subtracting polynomials to avoid errors in distribution.

What must be subtracted from 4x²y²+3xy+3xy² to obtain 8xy²-4x²y+7x²y²+10xy?

Learn how to solve a polynomial subtraction problem by finding the expression to subtract from 4x²y²+3xy+3xy² to obtain 8xy²-4x²y+7x²y²+10xy. This solution covers the core concepts of algebra, grouping like terms, and simplifying polynomials. Suitable for students in Grades 9-10.

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