What will be the sum of the coefficients of x square y cube and x square y in the given algerbraic expression (x square+y square)(6xsquare-y+4x square y)

To find the sum of the coefficients of the terms \( x^2y^3 \) and \( x^2y \) in the given algebraic expression \((x^2 + y^2)(6x^2 - y + 4x^2y)\), we need to expand the expression and then identify the required terms.

### Step 1: Expand the expression
\[
(x^2 + y^2)(6x^2 - y + 4x^2y)
\]

Distribute each term in the first parenthesis over the second parenthesis:

\[
= x^2(6x^2 - y + 4x^2y) + y^2(6x^2 - y + 4x^2y)
\]

Now, expand each term:

\[
= x^2 \cdot 6x^2 + x^2 \cdot (-y) + x^2 \cdot 4x^2y + y^2 \cdot 6x^2 + y^2 \cdot (-y) + y^2 \cdot 4x^2y
\]

\[
= 6x^4 + (-x^2y) + 4x^4y + 6x^2y^2 + (-y^3) + 4x^2y^3
\]

### Step 2: Identify the coefficients of \( x^2y^3 \) and \( x^2y \)
- The term involving \( x^2y^3 \) is \( 4x^2y^3 \), with a coefficient of \( 4 \).
- The term involving \( x^2y \) is \( -x^2y \), with a coefficient of \( -1 \).

### Step 3: Sum the coefficients
The sum of the coefficients is:
\[
4 + (-1) = 3
\]

### Final Answer
The sum of the coefficients of \( x^2y^3 \) and \( x^2y \) is **3**.

Would you like further details on the expansion process, or do you have any other questions?

Here are five related questions you might consider:
1. How do you expand the product of two binomials or polynomials?
2. What is the significance of the coefficient in a polynomial term?
3. How do you identify like terms in a polynomial?
4. What methods can be used to simplify polynomial expressions?
5. How do you solve polynomial equations by factoring?

**Tip:** When expanding polynomials, keep track of each term's degree and coefficients to avoid missing or incorrectly combining terms.

Learn how to find the sum of coefficients of x^2y^3 and x^2y in the algebraic expression (x^2 + y^2)(6x^2 - y + 4x^2y). This problem involves polynomial expansion and coefficient identification. Suitable for high school students studying algebra and polynomials.

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