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!