Let's simplify and solve the given expression:

$-x(8x - 3)(2x + 5)^2 - 0$

We can break this down step by step:

Step 1: Simplify $(2x + 5)^2$

First, expand the square:

$(2x + 5)^2 = (2x + 5)(2x + 5)$ Using the distributive property (FOIL method):

$= 4x^2 + 10x + 10x + 25 = 4x^2 + 20x + 25$

Step 2: Substitute $(2x + 5)^2$ back into the original expression

Now, the expression becomes:

$-x(8x - 3)(4x^2 + 20x + 25)$

Step 3: Expand the product

Next, we distribute $-(x)$ across $(8x - 3)$ and then multiply the result by $(4x^2 + 20x + 25)$.

First, expand $-(x)(8x - 3)$:

$-x(8x - 3) = -8x^2 + 3x$

Now, multiply this by $(4x^2 + 20x + 25)$:

$(-8x^2 + 3x)(4x^2 + 20x + 25)$

Step 4: Apply distributive property again

Distribute each term of $-8x^2 + 3x$ to $(4x^2 + 20x + 25)$:

$-8x^2 \cdot (4x^2 + 20x + 25) + 3x \cdot (4x^2 + 20x + 25)$

• For the first part:
  $-8x^2(4x^2 + 20x + 25) = -32x^4 - 160x^3 - 200x^2$

• For the second part:
  $3x(4x^2 + 20x + 25) = 12x^3 + 60x^2 + 75x$

Step 5: Combine the results

Now combine all the terms:

$-32x^4 - 160x^3 - 200x^2 + 12x^3 + 60x^2 + 75x$

Combine like terms:

$= -32x^4 + (-160x^3 + 12x^3) + (-200x^2 + 60x^2) + 75x$

Simplifying:

$= -32x^4 - 148x^3 - 140x^2 + 75x$

Final Expression

The simplified expression is:

$-32x^4 - 148x^3 - 140x^2 + 75x$

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Would you like to solve this for specific values of $x$, or do you have any other questions? Here are some related questions to explore:

1. How do you factor the expression $-32x^4 - 148x^3 - 140x^2 + 75x$?
2. How would you find the derivative of the simplified polynomial?
3. What is the behavior of the function as $x \to \infty$?
4. How would you solve for the roots of this polynomial?
5. How can you apply the distributive property for polynomials with higher degrees?

Tip: When simplifying polynomials, always try to combine like terms first to make your work easier.