## Factoring and Simplifying the Expression: 0.5x(2x^2-5)(2x^2+5)

This expression involves a combination of multiplication and factoring. Let's break it down step by step to understand how to simplify it.

### Understanding the Structure

The expression is a product of three factors:

**0.5x:**This is a simple linear term.**(2x^2 - 5):**This is a difference of squares pattern.**(2x^2 + 5):**This is also a difference of squares pattern.

### Applying the Difference of Squares Pattern

The difference of squares pattern states that: **a^2 - b^2 = (a + b)(a - b)**

Using this pattern, we can factor both (2x^2 - 5) and (2x^2 + 5):

**(2x^2 - 5) = (√2x + √5)(√2x - √5)****(2x^2 + 5) = (√2x + √5)(√2x - √5)**

### Simplifying the Expression

Now, let's substitute these factored forms back into the original expression:

0.5x(2x^2 - 5)(2x^2 + 5) = 0.5x (√2x + √5)(√2x - √5) (√2x + √5)(√2x - √5)

Notice that we have two identical pairs of factors: (√2x + √5) and (√2x - √5). This allows us to simplify further:

0.5x (√2x + √5)(√2x - √5) (√2x + √5)(√2x - √5) = 0.5x [(√2x + √5)(√2x - √5)]^2

Applying the difference of squares pattern again:

0.5x [(√2x + √5)(√2x - √5)]^2 = 0.5x [(2x^2 - 5)]^2

### Expanding and Final Result

Finally, we can expand the square:

0.5x [(2x^2 - 5)]^2 = 0.5x (4x^4 - 20x^2 + 25)

This is the simplified form of the original expression.

**Therefore, 0.5x(2x^2-5)(2x^2+5) is equivalent to 0.5x(4x^4 - 20x^2 + 25).**