Solve the equation:
`(2x-7)(x^(2)-9)(2x+5)=91`.

Answer

To solve the equation \((2x-7)(x^2-9)(2x+5) = 91\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ (2x-7)(x^2-9)(2x+5) = 91 \]
We can rewrite \(x^2 - 9\) as \((x-3)(x+3)\) because it is a difference of squares.

### Step 2: Substitute the expression
Substituting this back into the equation gives us:
\[ (2x-7)(x-3)(x+3)(2x+5) = 91 \]

### Step 3: Move 91 to the left side
Now, we can rewrite the equation as:
\[ (2x-7)(x-3)(x+3)(2x+5) - 91 = 0 \]

### Step 4: Let \(t = 2x^2 - x\)
To simplify the multiplication, we can let:
\[ t = 2x^2 - x \]
So we will express our equation in terms of \(t\).

### Step 5: Expand the left-hand side
Now we will expand the left-hand side:
\[ (2x-7)(x^2-9)(2x+5) = 0 \]
This leads to:
\[ (2x-7)(x^2-9)(2x+5) = (2x-7)(x-3)(x+3)(2x+5) \]

### Step 6: Expand and simplify
We can expand \((2x-7)(2x+5)\) and \((x-3)(x+3)\):
1. \((x-3)(x+3) = x^2 - 9\)
2. \((2x-7)(2x+5) = 4x^2 - 14x + 10x - 35 = 4x^2 - 4x - 35\)

Now we have:
\[ (4x^2 - 4x - 35)(x^2 - 9) = 91 \]

### Step 7: Set up the quadratic equation
Expanding this gives us a polynomial that we can set equal to zero:
\[ 4x^4 - 36x^2 - 4x^3 + 36x - 35 + 91 = 0 \]
This simplifies to:
\[ 4x^4 - 4x^3 - 36x^2 + 36x + 56 = 0 \]

### Step 8: Factor the polynomial
We can factor this polynomial. Let's look for rational roots or use synthetic division to find factors.

### Step 9: Solve for \(x\)
After factoring, we will have:
\[ (2x^2 - 7)(2x^2 + 8) = 0 \]
Setting each factor to zero gives us:
1. \(2x^2 - 7 = 0 \Rightarrow x^2 = \frac{7}{2} \Rightarrow x = \pm \sqrt{\frac{7}{2}}\)
2. \(2x^2 + 8 = 0\) has no real solutions.

### Step 10: Final solutions
Thus, the real solutions for \(x\) are:
\[ x = \sqrt{\frac{7}{2}}, \quad x = -\sqrt{\frac{7}{2}} \]