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The problem asks for the product of the expressions \(\frac{1}{2}x - \frac{1}{4}\) and \(5x^2 - 2x + 6\), with the result written in standard form.

Let's work through this step by step:

### Step 1: Distribute \(\frac{1}{2}x\) to each term in the trinomial \(5x^2 - 2x + 6\).
\[
\frac{1}{2}x \cdot (5x^2 - 2x + 6) = \left(\frac{1}{2}x \cdot 5x^2\right) + \left(\frac{1}{2}x \cdot (-2x)\right) + \left(\frac{1}{2}x \cdot 6\right)
\]
1. \(\frac{1}{2}x \cdot 5x^2 = \frac{5}{2}x^3\)
2. \(\frac{1}{2}x \cdot (-2x) = -x^2\)
3. \(\frac{1}{2}x \cdot 6 = 3x\)

So the result of distributing \(\frac{1}{2}x\) is:
\[
\frac{5}{2}x^3 - x^2 + 3x
\]

### Step 2: Distribute \(-\frac{1}{4}\) to each term in the trinomial \(5x^2 - 2x + 6\).
\[
-\frac{1}{4} \cdot (5x^2 - 2x + 6) = \left(-\frac{1}{4} \cdot 5x^2\right) + \left(-\frac{1}{4} \cdot (-2x)\right) + \left(-\frac{1}{4} \cdot 6\right)
\]
1. \(-\frac{1}{4} \cdot 5x^2 = -\frac{5}{4}x^2\)
2. \(-\frac{1}{4} \cdot (-2x) = \frac{1}{2}x\)
3. \(-\frac{1}{4} \cdot 6 = -\frac{3}{2}\)

So the result of distributing \(-\frac{1}{4}\) is:
\[
-\frac{5}{4}x^2 + \frac{1}{2}x - \frac{3}{2}
\]

### Step 3: Combine like terms.
Now add the two results from Step 1 and Step 2:
\[
\left(\frac{5}{2}x^3 - x^2 + 3x\right) + \left(-\frac{5}{4}x^2 + \frac{1}{2}x - \frac{3}{2}\right)
\]

Combine the \(x^2\) terms:
\[
-x^2 - \frac{5}{4}x^2 = -\frac{4}{4}x^2 - \frac{5}{4}x^2 = -\frac{9}{4}x^2
\]

Combine the \(x\) terms:
\[
3x + \frac{1}{2}x = \frac{6}{2}x + \frac{1}{2}x = \frac{7}{2}x
\]

So the full expression is:
\[
\frac{5}{2}x^3 - \frac{9}{4}x^2 + \frac{7}{2}x - \frac{3}{2}
\]

### Final Answer (in standard form):
\[
\boxed{\frac{5}{2}x^3 - \frac{9}{4}x^2 + \frac{7}{2}x - \frac{3}{2}}
\]

Would you like further details or have any questions?

Here are some related questions you might want to explore:
1. How do you simplify algebraic expressions involving fractions?
2. What is the standard form of a polynomial?
3. How do you multiply binomials and trinomials?
4. Can you solve a polynomial equation using the distributive property?
5. What are some real-world applications of polynomial multiplication?

**Tip:** Always group like terms after expanding expressions to ensure the answer is in its simplest form.

What is the product of \(\frac{1}{2}x - \frac{1}{4}\) and \(5x^2 - 2x + 6\)? Write your answer in standard form.

Learn how to find the product of \(\frac{1}{2}x - \frac{1}{4}\) and \(5x^2 - 2x + 6\) with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomial multiplication and the distributive property. Suitable for students in Grades 9-12.

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