https://r2.erweima.ai/mathbot/image/172719767525862pbvnvb.jpg

The image contains two algebraic expressions that need to be simplified or reduced. Let's break down each expression and simplify it.

### Expression 1:
\[
x^2 - 7y^2 + 12z^2 - 9x^2 + 6y^2 - 4z^2
\]

To simplify this expression, we combine like terms:

- For \(x^2\) terms: \(x^2 - 9x^2 = -8x^2\)
- For \(y^2\) terms: \(-7y^2 + 6y^2 = -y^2\)
- For \(z^2\) terms: \(12z^2 - 4z^2 = 8z^2\)

Thus, the simplified form is:
\[
-8x^2 - y^2 + 8z^2
\]

### Expression 2:
\[
-7m^6 + 4n^4 - 11p^5 - 5m^6 + 8p^5 + 6n^4 - n^4 + 14m^6 - 2p^5
\]

Similarly, we combine like terms:

- For \(m^6\) terms: \(-7m^6 - 5m^6 + 14m^6 = 2m^6\)
- For \(n^4\) terms: \(4n^4 + 6n^4 - n^4 = 9n^4\)
- For \(p^5\) terms: \(-11p^5 + 8p^5 - 2p^5 = -5p^5\)

Thus, the simplified form is:
\[
2m^6 + 9n^4 - 5p^5
\]

Would you like further clarification or any additional details?

### Related Questions:
1. How do you combine like terms in polynomial expressions?
2. What are the general rules for simplifying algebraic expressions?
3. Can polynomials have different degrees for each term?
4. What are some common mistakes to avoid when simplifying algebraic expressions?
5. How would you factor a quadratic expression after simplifying?

### Tip:
Always group terms with the same variable and exponent when simplifying algebraic expressions to avoid mistakes.

Suma, simplifica o reduce las siguientes expresiones: x^2 - 7y^2 + 12z^2 - 9x^2 + 6y^2 - 4z^2 y -7m^6 + 4n^4 - 11p^5 - 5m^6 + 8p^5 + 6n^4 - n^4 + 14m^6 - 2p^5

Learn how to simplify algebraic expressions like x^2 - 7y^2 + 12z^2 - 9x^2 + 6y^2 - 4z^2 and -7m^6 + 4n^4 - 11p^5. This lesson focuses on combining like terms in polynomials. Suitable for Grades 9-10.

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