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The inequality given is:

\[
2(x - 4)(x + 4) > 3x(2x + 3)
\]

Let's solve this inequality step by step:

1. **Expand both sides:**

   Left side:  
   \[
   2(x - 4)(x + 4) = 2(x^2 - 16) = 2x^2 - 32
   \]

   Right side:  
   \[
   3x(2x + 3) = 6x^2 + 9x
   \]

2. **Rewrite the inequality:**

   \[
   2x^2 - 32 > 6x^2 + 9x
   \]

3. **Move all terms to one side:**

   \[
   2x^2 - 6x^2 - 9x - 32 > 0
   \]

   Simplify:

   \[
   -4x^2 - 9x - 32 > 0
   \]

   To make it more conventional, multiply by \(-1\):

   \[
   4x^2 + 9x + 32 < 0
   \]

4. **Analyze the quadratic inequality:**

   For a quadratic \(ax^2 + bx + c < 0\), we need to find the roots using the discriminant:

   Discriminant (\(\Delta\)) is given by:

   \[
   \Delta = b^2 - 4ac = 9^2 - 4(4)(32) = 81 - 512 = -431
   \]

Since the discriminant is negative (\(\Delta < 0\)), the quadratic equation has no real roots and is always positive since \(a > 0\) for \(4x^2 + 9x + 32\). 

Therefore, \(4x^2 + 9x + 32 < 0\) has no real solutions.

**Conclusion:** The inequality \(2(x - 4)(x + 4) > 3x(2x + 3)\) holds for no real values of \(x\).

Would you like further details or have any specific questions?

Here are some related questions you might consider:

1. How can inequalities be solved graphically?
2. What are the implications of a negative discriminant in a quadratic equation?
3. How does one solve inequalities involving higher-order polynomials?
4. What techniques can be used to analyze the behavior of quadratic functions?
5. How do you determine the intervals of positivity and negativity of a quadratic function?

**Tip:** When solving inequalities, always check the nature of the quadratic (positive or negative discriminant) to determine if the expression can ever be less than or greater than zero.

Solve the inequality 2(x - 4)(x + 4) > 3x(2x + 3)

Learn how to solve the quadratic inequality 2(x - 4)(x + 4) > 3x(2x + 3) with a step-by-step solution. Explore the expansion of quadratic terms, discriminant analysis, and the behavior of inequalities. Suitable for students in Grades 9-12.

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