(2x - 5)(2x + 5) - 3(x² - 7) - (8x⁴y³ + 8x³y³) ÷ (-4x³y³)

Let's solve the expression step by step:

### Expression:
\[
(2x - 5)(2x + 5) - 3(x^2 - 7) - \frac{8x^4y^3 + 8x^3y^3}{-4x^3y^3}
\]

### Step 1: Expand and simplify \((2x - 5)(2x + 5)\)
This is a difference of squares:
\[
(2x - 5)(2x + 5) = (2x)^2 - (5)^2 = 4x^2 - 25
\]

### Step 2: Distribute and simplify \(3(x^2 - 7)\)
\[
3(x^2 - 7) = 3x^2 - 21
\]

### Step 3: Simplify the division \(\frac{8x^4y^3 + 8x^3y^3}{-4x^3y^3}\)
Factor out \(8x^3y^3\) from the numerator:
\[
\frac{8x^3y^3(x + 1)}{-4x^3y^3} = -2(x + 1)
\]

### Step 4: Combine all the simplified expressions
Now substitute back:
\[
(4x^2 - 25) - (3x^2 - 21) - [-2(x + 1)]
\]
Simplify further:
\[
4x^2 - 25 - 3x^2 + 21 + 2(x + 1)
\]

### Step 5: Combine like terms
\[
(4x^2 - 3x^2) + (-25 + 21) + (2x + 2)
\]
\[
x^2 - 4 + 2x + 2
\]

Finally, combine the constants:
\[
x^2 + 2x - 2
\]

### Final Answer:
\[
x^2 + 2x - 2
\]

Would you like details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you factor the expression \(x^2 + 2x - 2\)?
2. What are the roots of the quadratic equation \(x^2 + 2x - 2 = 0\)?
3. How does the distributive property work when expanding polynomials?
4. What is the difference between a polynomial of degree 2 and degree 4?
5. How does the sign change in a fraction when the denominator is negative?

**Tip:** When dealing with polynomial expressions, always check for common factors to simplify the expression quickly.

Learn how to solve the polynomial expression (2x - 5)(2x + 5) - 3(x^2 - 7) - (8x^4y^3 + 8x^3y^3) / (-4x^3y^3) step by step with detailed explanations and examples. This problem covers key algebraic concepts such as polynomials, quadratic equations, simplification, and factoring. Suitable for Grades 9-12 students and beyond.

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