https://file.aiquickdraw.com/mathbot/file/1734986782362z4z3yuon.jpg

Here are the mathematical expressions from the image, along with solutions to simplify each:

### 12. \( 2.3x^2 + 3 - 4.1x^2 + 3x \)
Combine like terms:
- \( x^2 \)-terms: \( 2.3x^2 - 4.1x^2 = -1.8x^2 \)
- Constant terms: \( 3 \)
- \( x \)-terms: \( 3x \)

Simplified expression:  
\[
-1.8x^2 + 3x + 3
\]

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### 13. \( 3y^4 - 2y^2 - y^4 - 2y \)
Combine like terms:
- \( y^4 \)-terms: \( 3y^4 - y^4 = 2y^4 \)
- \( y^2 \)-terms: \( -2y^2 \)
- \( y \)-terms: \( -2y \)

Simplified expression:  
\[
2y^4 - 2y^2 - 2y
\]

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### 14. \( -2x^3 - 2x^2 - 2x^3 + 2x^2 \)
Combine like terms:
- \( x^3 \)-terms: \( -2x^3 - 2x^3 = -4x^3 \)
- \( x^2 \)-terms: \( -2x^2 + 2x^2 = 0 \)

Simplified expression:  
\[
-4x^3
\]

---

### 15. \( x^2 - 2x + x^3 + x \)
Rearrange terms in descending powers of \( x \):
\[
x^3 + x^2 - 2x + x
\]

Combine like terms:
- \( x \)-terms: \( -2x + x = -x \)

Simplified expression:  
\[
x^3 + x^2 - x
\]

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Would you like further details or explanations for any of the steps?  

Here are 5 follow-up questions to explore further:  
1. What methods can we use to factorize these simplified expressions?  
2. How do the coefficients influence the shape of the polynomial graph?  
3. Can you identify the degree of each polynomial and explain its significance?  
4. How would you evaluate each polynomial for a specific value of the variable?  
5. How does combining like terms work when dealing with higher powers of variables?

**Tip:** Always arrange terms in descending order of their degree for easier manipulation and readability!

Simplify the following polynomial expressions as provided in the image.

Explore step-by-step solutions to simplify polynomial expressions, including terms with powers of x and y. Understand key algebra concepts such as combining like terms and arranging terms in descending order. Suitable for Grades 8-10.

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