M³-m2+m-2 por am+a
5m⁴-3m²n²+n⁴ por 3m-n
X²-4x+3 por -2x
a+b+c+d restar -a-b-c-d
8a+b restar -3a+4 
(7x+2y-4)+(9y-6z+5)+(-y+3z-6)+(-5+8x-3y)

Let's break down each expression step by step:

### 1. \( M^3 - M^2 + M - 2 \) multiplied by \( aM + a \)
We distribute \( (aM + a) \) to each term in \( M^3 - M^2 + M - 2 \).

\[
(aM + a) \cdot (M^3 - M^2 + M - 2) = aM \cdot (M^3 - M^2 + M - 2) + a \cdot (M^3 - M^2 + M - 2)
\]
\[
= aM \cdot M^3 - aM \cdot M^2 + aM \cdot M - 2aM + a \cdot M^3 - a \cdot M^2 + a \cdot M - 2a
\]
\[
= aM^4 - aM^3 + aM^2 - 2aM + aM^3 - aM^2 + aM - 2a
\]
\[
= aM^4 - aM^2 + aM - 2a
\]
Final answer:
\[
aM^4 + aM^3 - aM^2 - 2aM - 2a
\]

---

### 2. \( 5M^4 - 3M^2N^2 + N^4 \) multiplied by \( 3M - N \)
We distribute \( (3M - N) \) to each term in \( 5M^4 - 3M^2N^2 + N^4 \).

\[
(3M - N) \cdot (5M^4 - 3M^2N^2 + N^4) = 3M \cdot (5M^4 - 3M^2N^2 + N^4) - N \cdot (5M^4 - 3M^2N^2 + N^4)
\]
\[
= 15M^5 - 9M^3N^2 + 3M \cdot N^4 - 5M^4N + 3M^2N^3 - N^5
\]
Final answer:
\[
15M^5 - 5M^4N - 9M^3N^2 + 3M^2N^3 + 3MN^4 - N^5
\]

---

### 3. \( X^2 - 4X + 3 \) multiplied by \( -2X \)
We distribute \( -2X \) to each term in \( X^2 - 4X + 3 \).

\[
-2X \cdot (X^2 - 4X + 3) = -2X^3 + 8X^2 - 6X
\]
Final answer:
\[
-2X^3 + 8X^2 - 6X
\]

---

### 4. \( a + b + c + d \) minus \( -a - b - c - d \)
This simplifies by changing the signs of the second expression:
\[
(a + b + c + d) - (-a - b - c - d) = a + b + c + d + a + b + c + d
\]
\[
= 2a + 2b + 2c + 2d
\]
Final answer:
\[
2a + 2b + 2c + 2d
\]

---

### 5. \( 8a + b \) minus \( -3a + 4 \)
This simplifies by changing the signs of the second expression:
\[
(8a + b) - (-3a + 4) = 8a + b + 3a - 4
\]
\[
= 11a + b - 4
\]
Final answer:
\[
11a + b - 4
\]

---

### 6. \( (7x + 2y - 4) + (9y - 6z + 5) + (-y + 3z - 6) + (-5 + 8x - 3y) \)
Combine like terms:
- \( x \)-terms: \( 7x + 8x = 15x \)
- \( y \)-terms: \( 2y + 9y - y - 3y = 7y \)
- \( z \)-terms: \( -6z + 3z = -3z \)
- Constant terms: \( -4 + 5 - 6 - 5 = -10 \)

Final answer:
\[
15x + 7y - 3z - 10
\]

---

Do you want any further explanation or more details on any of the steps?

Here are 5 follow-up questions you can explore:
1. How can I multiply polynomials using the distributive property more efficiently?
2. How do you simplify expressions involving both addition and subtraction of terms?
3. What strategies help identify like terms in complex expressions?
4. How does distributing negative terms affect polynomial multiplication?
5. Can I use factoring to simplify polynomial expressions before multiplication?

**Tip:** Always look for like terms when simplifying or combining expressions—they help reduce complexity and minimize errors!

M³-m²+m-2 por am+a, 5m⁴-3m²n²+n⁴ por 3m-n, X²-4x+3 por -2x, a+b+c+d restar -a-b-c-d, 8a+b restar -3a+4, (7x+2y-4)+(9y-6z+5)+(-y+3z-6)+(-5+8x-3y)

Explore detailed solutions for multiplying and simplifying polynomials, such as M³ - M² + M - 2 multiplied by am + a, or subtracting polynomial expressions like 8a + b and -3a + 4. This page covers polynomial multiplication, distributive property, and combining like terms, ideal for students in Grades 9-12.

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