Simplify (a^2-ab+1/2b^2) (3/4a^2+2ab-b^2)

To simplify the expression \((a^2 - ab + \frac{1}{2}b^2)(\frac{3}{4}a^2 + 2ab - b^2)\), we will expand it using the distributive property (also known as FOIL for binomials). 

### Step 1: Expand the expression
We multiply each term in the first parenthesis by each term in the second parenthesis.

\[
(a^2 - ab + \frac{1}{2}b^2) \cdot \left(\frac{3}{4}a^2 + 2ab - b^2\right)
\]

#### First, multiply \(a^2\) by each term in the second expression:

\[
a^2 \cdot \frac{3}{4}a^2 = \frac{3}{4}a^4
\]
\[
a^2 \cdot 2ab = 2a^3b
\]
\[
a^2 \cdot (-b^2) = -a^2b^2
\]

#### Second, multiply \(-ab\) by each term in the second expression:

\[
-ab \cdot \frac{3}{4}a^2 = -\frac{3}{4}a^3b
\]
\[
-ab \cdot 2ab = -2a^2b^2
\]
\[
-ab \cdot (-b^2) = ab^3
\]

#### Third, multiply \(\frac{1}{2}b^2\) by each term in the second expression:

\[
\frac{1}{2}b^2 \cdot \frac{3}{4}a^2 = \frac{3}{8}a^2b^2
\]
\[
\frac{1}{2}b^2 \cdot 2ab = ab^3
\]
\[
\frac{1}{2}b^2 \cdot (-b^2) = -\frac{1}{2}b^4
\]

### Step 2: Combine all the terms
Now, we sum all the terms:

\[
\frac{3}{4}a^4 + 2a^3b - a^2b^2 - \frac{3}{4}a^3b - 2a^2b^2 + ab^3 + \frac{3}{8}a^2b^2 + ab^3 - \frac{1}{2}b^4
\]

### Step 3: Simplify
Group the like terms together:

- \(a^4\) term: \(\frac{3}{4}a^4\)
- \(a^3b\) terms: \(2a^3b - \frac{3}{4}a^3b = \frac{5}{4}a^3b\)
- \(a^2b^2\) terms: \(-a^2b^2 - 2a^2b^2 + \frac{3}{8}a^2b^2 = -\frac{25}{8}a^2b^2\)
- \(ab^3\) terms: \(ab^3 + ab^3 = 2ab^3\)
- \(b^4\) term: \(-\frac{1}{2}b^4\)

Thus, the simplified expression is:

\[
\boxed{\frac{3}{4}a^4 + \frac{5}{4}a^3b - \frac{25}{8}a^2b^2 + 2ab^3 - \frac{1}{2}b^4}
\]

Would you like more details or have any questions?

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Here are 5 related questions to explore further:
1. How do you expand the product of two binomials?
2. What happens when two like terms have fractional coefficients?
3. How do you factor a quadratic expression involving two variables?
4. Can you simplify this expression using a different method, like factoring?
5. What is the degree of each term in this expanded expression?

**Tip**: Always group like terms carefully, paying attention to signs and coefficients!

Learn how to simplify the polynomial expression (a^2 - ab + 1/2b^2)(3/4a^2 + 2ab - b^2) using the distributive property. This problem covers polynomial expansion, combining like terms, and algebraic simplification. Suitable for high school students in Grades 10-12.

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