Solving the Equation: (x^2  3x/2 + 3)(x^2  3x/2  4) + 10 = 0
This equation presents a challenge due to its complex structure. Here's a stepbystep approach to solve it:
1. Simplifying the Expression
First, let's make the equation easier to work with by simplifying the expression:

Let y = x^2  3x/2 This substitution allows us to rewrite the equation as: (y + 3)(y  4) + 10 = 0

Expanding the Equation: y^2  y  12 + 10 = 0 y^2  y  2 = 0
2. Solving the Quadratic Equation
Now we have a standard quadratic equation in terms of 'y':
 Factoring: (y  2)(y + 1) = 0 Therefore, y = 2 or y = 1
3. Substituting Back
We need to substitute back 'x' for 'y' to find the solutions:

For y = 2: x^2  3x/2 = 2 2x^2  3x  4 = 0 This quadratic equation can be solved using the quadratic formula.

For y = 1: x^2  3x/2 = 1 2x^2  3x + 2 = 0 This quadratic equation can also be solved using the quadratic formula.
4. Finding the Solutions
The quadratic formula will provide the solutions for both equations obtained in step 3. Remember to calculate both possible values for 'x' in each equation.
Therefore, the solutions to the original equation (x^2  3x/2 + 3)(x^2  3x/2  4) + 10 = 0 are the roots obtained from solving the two quadratic equations derived in step 3.
Note: The quadratic formula provides two possible solutions for each quadratic equation. It's essential to verify if all these solutions satisfy the original equation.