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

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.

Learn how to simplify the polynomial expression (a^2 - ab + 1/2b^2)(3/4a^2 + 2ab - b^2) using the distributive property. This detailed step-by-step solution covers key algebra concepts and is suitable for students in Grades 9-12.

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