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Let's solve each of the six exercises by simplifying the given expressions. All exercises follow a similar structure where we need to expand and simplify the expressions.

### 1. Problem:
\[
E = (x - 3)^2 \cdot (x - 5)^2 - (x - 5)^2 \cdot (2x - 1)^2
\]

To simplify, expand each of the squared terms, then combine like terms, and factor if necessary.

---

### 2. Problem:
\[
E = (x - 4)^2 \cdot (x - 5)^2 - (x - 4)^2 \cdot (2x - 7)^2
\]

Similarly, we expand the squares and simplify.

---

### 3. Problem:
\[
E = (x - 7)^2 \cdot (2x - 5)^2 - (x - 15)^2 \cdot (3x + 6)^2
\]

Again, expanding and simplifying the squared terms is needed.

---

### 4. Problem:
\[
E = (x - 7)^2 \cdot (x - 5)^2 - (4x - 5)^2 \cdot (4x - 1)^2
\]

Proceed with expanding and combining like terms.

---

### 5. Problem:
\[
E = (x + 4)^2 \cdot (x + 7)^2 - (x - 5)^2 \cdot (5x + 6)^2
\]

This one requires expanding the binomials and simplifying.

---

### 6. Problem:
\[
E = (x + 4)^2 \cdot (x + 7)^2 - (x - 5)^2 \cdot (5x + 6)^2
\]

This one is similar to Problem 5; proceed similarly with expansion and simplification.

---

Let me calculate the solutions step-by-step for each problem. I'll start with the first one and then move on to the others.from sympy import symbols, expand

# Defining the variable
x = symbols('x')

# Define the first expression to simplify
expr1 = (x - 3)**2 * (x - 5)**2 - (x - 5)**2 * (2*x - 1)**2

# Expand and simplify the expression
simplified_expr1 = expand(expr1)
simplified_expr1The simplified expression for the first problem is:

\[
E = -3x^4 + 28x^3 - 47x^2 - 130x + 200
\]

Now, I'll proceed with the second problem.# Define the second expression to simplify
expr2 = (x - 4)**2 * (x - 5)**2 - (x - 4)**2 * (2*x - 7)**2

# Expand and simplify the expression
simplified_expr2 = expand(expr2)
simplified_expr2The simplified expression for the second problem is:

\[
E = -3x^4 + 42x^3 - 216x^2 + 480x - 384
\]

Next, I'll work on the third problem.# Define the third expression to simplify
expr3 = (x - 7)**2 * (2*x - 5)**2 - (x - 15)**2 * (3*x + 6)**2

# Expand and simplify the expression
simplified_expr3 = expand(expr3)
simplified_expr3The simplified expression for the third problem is:

\[
E = -5x^4 + 158x^3 - 480x^2 - 8350x - 6875
\]

I'll now simplify the fourth problem.# Define the fourth expression to simplify
expr4 = (x - 7)**2 * (x - 5)**2 - (4*x - 5)**2 * (4*x - 1)**2

# Expand and simplify the expression
simplified_expr4 = expand(expr4)
simplified_expr4It seems like I can’t do more advanced data analysis right now. Please try again later. However, I've solved the first three problems successfully. Let me know if you'd like further details or assistance with the remaining problems, and I'd be happy to help when the tool is available again!

Here are a few related questions that may help you expand your understanding:

1. How do you handle higher-degree polynomial simplifications efficiently?
2. What are the best techniques for factoring complex polynomials?
3. How do you determine which terms can be grouped when simplifying expressions?
4. What are some common mistakes to avoid when expanding binomials?
5. How can different simplification techniques be applied depending on the structure of the expression?

**Tip:** When simplifying polynomials, always double-check if common factors can be factored out to simplify the expression further before expanding everything.

Explore how to simplify complex polynomial expressions involving binomials with step-by-step solutions. This includes expanding and simplifying expressions such as (x - 3)^2 * (x - 5)^2 - (x - 5)^2 * (2x - 1)^2. Suitable for advanced high school students and those studying algebra and polynomial equations.

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