https://file.aiquickdraw.com/mathbot/file/1725259019411v5fttq5q.jpg

The expression given in the image is:

\[
\frac{\sqrt{6}}{3} - 40x^2
\]

To simplify and factor this expression step by step:

### Step 1: Simplify the Fraction
The first term, \(\frac{\sqrt{6}}{3}\), is already in its simplest form. There's no further simplification for this fraction.

### Step 2: Factor the Polynomial Term
The second term, \(-40x^2\), is a quadratic term that can be factored by finding common factors. 

Here, we can factor out a common factor of 40 from the second term, but since the term is negative, we'll factor out \(-40\) as a common factor:

\[
-40x^2 = -40(x^2)
\]

### Final Expression:
The expression after simplifying and factoring becomes:

\[
\frac{\sqrt{6}}{3} - 40(x^2)
\]

Since the two terms don't share any common factors or expressions that can be factored further, this is the simplified form.

### Recap:
1. Simplified the fraction \(\frac{\sqrt{6}}{3}\).
2. Factored out \(-40\) from \(-40x^2\).

This is the fully simplified form. If you need further details or if you have any questions, feel free to ask!

---

### Follow-up Questions:
1. How would you factor \(\frac{\sqrt{6}}{3} - 40x^2\) if the expression were more complex?
2. Can you find the value of the expression for a specific value of \(x\)?
3. What happens to the expression as \(x\) approaches infinity?
4. How would the expression change if the term were \(+40x^2\) instead of \(-40x^2\)?
5. How can you use this expression in a real-world problem?

### Tip:
When simplifying expressions, always look for common factors or special identities like the difference of squares. This can help in factoring complex expressions more easily.

Learn how to simplify the expression \(\frac{\sqrt{6}}{3} - 40x^2\) step by step by simplifying fractions and factoring polynomials. This problem involves simplifying a fraction and factoring a quadratic polynomial term. Suitable for students in Grades 7-9.

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