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The image contains a set of mathematical expressions to simplify. I'll transcribe the expressions and solve them one by one.

### 1. Simplify the following algebraic expressions.

**a)** \((6x^2 - 1)(5x^2 - 6x - 1)\)

**b)** \((5x^3 + 2x + 5)(4x^2 - 2x - 1)\)

**c)** \((3x + 4 - 8)(2x^2 - x + 5)\)

**d)** \((x^4 - 4x - 1) - (3x^2 - 8x + 5)\)

**e)** \((x^2 + 2x - 7) + (4x^2 - 2x + 6)\)

I will simplify each of these expressions step by step.

### Solution to **a**: 
\[
(6x^2 - 1)(5x^2 - 6x - 1)
\]
Expanding the expression using the distributive property (FOIL method for binomials):
\[
6x^2(5x^2) + 6x^2(-6x) + 6x^2(-1) - 1(5x^2) - 1(-6x) - 1(-1)
\]
Simplifying the terms:
\[
30x^4 - 36x^3 - 6x^2 - 5x^2 + 6x + 1
\]
Combining like terms:
\[
30x^4 - 36x^3 - 11x^2 + 6x + 1
\]

Let me now work on the other parts.The simplified form of expression **a** is:
\[
30x^4 - 36x^3 - 11x^2 + 6x + 1
\]

I will now continue simplifying the other expressions. Let's move on to **b**:

\[
(5x^3 + 2x + 5)(4x^2 - 2x - 1)
\]
I'll expand this expression next.The simplified form of expression **b** is:
\[
20x^5 - 10x^4 + 3x^3 + 16x^2 - 12x - 5
\]

Next, I'll simplify expression **c**:
\[
(3x + 4 - 8)(2x^2 - x + 5)
\]
which simplifies to:
\[
(3x - 4)(2x^2 - x + 5)
\]

Let me expand this now.The simplified form of expression **c** is:
\[
6x^3 - 11x^2 + 19x - 20
\]

Now, let's move on to simplifying expression **d**:
\[
(x^4 - 4x - 1) - (3x^2 - 8x + 5)
\]

I'll simplify this expression next.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can proceed by manually simplifying **d**:
\[
(x^4 - 4x - 1) - (3x^2 - 8x + 5)
\]
Distribute the negative sign:
\[
x^4 - 4x - 1 - 3x^2 + 8x - 5
\]
Simplify by combining like terms:
\[
x^4 - 3x^2 + 4x - 6
\]

I will now move on to **e**.

Expression **e** is:
\[
(x^2 + 2x - 7) + (4x^2 - 2x + 6)
\]
Let me simplify it manually:
\[
x^2 + 2x - 7 + 4x^2 - 2x + 6
\]
Combine like terms:
\[
5x^2 - 1
\]

Here are the simplified forms of the expressions:

- **a)** \(30x^4 - 36x^3 - 11x^2 + 6x + 1\)
- **b)** \(20x^5 - 10x^4 + 3x^3 + 16x^2 - 12x - 5\)
- **c)** \(6x^3 - 11x^2 + 19x - 20\)
- **d)** \(x^4 - 3x^2 + 4x - 6\)
- **e)** \(5x^2 - 1\)

Would you like further details or have any questions? Here are some related questions to explore:

1. How do you expand polynomials with more than two terms?
2. What is the FOIL method for binomial expansions?
3. How can you check if a polynomial factorization is correct?
4. What strategies can simplify polynomial subtraction?
5. How do you solve higher-degree polynomial equations?

**Tip**: Always double-check signs when distributing or subtracting terms to avoid common errors!

Simplify the following algebraic expressions: a) (6x^2 - 1)(5x^2 - 6x - 1), b) (5x^3 + 2x + 5)(4x^2 - 2x - 1), c) (3x + 4 - 8)(2x^2 - x + 5), d) (x^4 - 4x - 1) - (3x^2 - 8x + 5), e) (x^2 + 2x - 7) + (4x^2 - 2x + 6)

Simplify polynomial expressions using algebraic techniques like the distributive property and polynomial subtraction. This covers multiple cases of polynomial multiplication and simplification, suitable for Grades 8-10 students.

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