is (-ac+2bc+4c^2-3a^2-ab+2b^2) & (-3a^2-2b^2+4c^2-ab+6bc-7ac) same

To determine if the two expressions

\[
(-ac + 2bc + 4c^2 - 3a^2 - ab + 2b^2)
\]

and

\[
(-3a^2 - 2b^2 + 4c^2 - ab + 6bc - 7ac)
\]

are the same, we'll simplify both expressions and compare their like terms.

### Simplify the first expression:
\[
- ac + 2bc + 4c^2 - 3a^2 - ab + 2b^2
\]

Let's rewrite this expression by grouping like terms:
- For \( a^2 \): \( -3a^2 \)
- For \( b^2 \): \( +2b^2 \)
- For \( c^2 \): \( +4c^2 \)
- For \( ab \): \( -ab \)
- For \( ac \): \( -ac \)
- For \( bc \): \( +2bc \)

### Simplify the second expression:
\[
-3a^2 - 2b^2 + 4c^2 - ab + 6bc - 7ac
\]

Similarly, let's group like terms:
- For \( a^2 \): \( -3a^2 \)
- For \( b^2 \): \( -2b^2 \)
- For \( c^2 \): \( +4c^2 \)
- For \( ab \): \( -ab \)
- For \( ac \): \( -7ac \)
- For \( bc \): \( +6bc \)

### Compare the two simplified expressions:
- **\(a^2\) terms**: Both have \( -3a^2 \), so they are the same.
- **\(b^2\) terms**: The first expression has \( +2b^2 \) while the second has \( -2b^2 \). They differ here.
- **\(c^2\) terms**: Both have \( +4c^2 \), so they are the same.
- **\(ab\) terms**: Both have \( -ab \), so they are the same.
- **\(ac\) terms**: The first has \( -ac \), and the second has \( -7ac \). They differ here.
- **\(bc\) terms**: The first has \( +2bc \), and the second has \( +6bc \). They differ here.

### Conclusion:
The expressions are **not the same** because their coefficients for \(b^2\), \(ac\), and \(bc\) terms are different.

Would you like to explore why these differences occur? Here are some related questions:

1. How do you compare algebraic expressions effectively?
2. What are like terms in an algebraic expression, and why are they important?
3. How can differences in coefficients affect the equality of two algebraic expressions?
4. What techniques can be used to simplify complex algebraic expressions?
5. How do you determine the factorization of similar-looking algebraic expressions?

**Tip:** When comparing expressions, always align and compare like terms to ensure accuracy.

Explore whether the expressions (-ac + 2bc + 4c^2 - 3a^2 - ab + 2b^2) and (-3a^2 - 2b^2 + 4c^2 - ab + 6bc - 7ac) are identical through detailed simplification and comparison of like terms. Understand how coefficients affect the equality of algebraic expressions. Suitable for Grades 7-9.

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