Let's simplify the expression $(a^2 - ab + \frac{1}{2}b^2)(\frac{3}{4}a^2 + 2ab - b^2)$.

To do this, we'll apply the distributive property (also known as the FOIL method for binomials) and multiply each term in the first parenthesis by every term in the second parenthesis. Here’s the step-by-step process:

Given expression: $(a^2 - ab + \frac{1}{2}b^2)(\frac{3}{4}a^2 + 2ab - b^2)$

Step 1: Multiply $a^2$ by every term in $\frac{3}{4}a^2 + 2ab - b^2$

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

Step 2: Multiply $-ab$ by every term in $\frac{3}{4}a^2 + 2ab - b^2$

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

Step 3: Multiply $\frac{1}{2}b^2$ by every term in $\frac{3}{4}a^2 + 2ab - b^2$

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

Step 4: Combine all the terms

Now we collect 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 5: Simplify the terms

• $a^4$ terms: $\frac{3}{4}a^4$
• $a^3b$ 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: $-\frac{1}{2}b^4$

Final simplified expression:

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

This is the simplified form of the given expression.

Would you like further details or explanations on any step? Here are five related questions to practice:

1. Simplify $(a^2 - 3ab + b^2)(2a - b)$.
2. Expand $(2x^2 - xy + y^2)(x + y)$.
3. Factor $a^4 + 5a^3b - 25a^2b^2 + 16ab^3 - b^4$.
4. Simplify $(x^2 + y)(x^3 - 2xy + y^2)$.
5. Expand and simplify $(2a + b)(a^2 - ab + b^2)$.

Tip: When simplifying expressions involving polynomials, always look for like terms to combine after expansion.