Use factoring by grouping to solve the following equation.

4x3 + 8x2 − 9x − 18 = 0
x = 1 (smallest value)
x = 2
x = 3 (largest value)

x = 1 is not a solution. You can easily prove that yourself.

When x = 2,
4x^3 + 8x^2 − 9x − 18 = 28

So that is not a solution, either. Neither is x = 3. How did you come up with your answers?

When x = -2,
-32 +32 +18 -18 = 0

So x+2 is a factor of the polynomial.
The other factor is
(4x^3 + 8x^2 − 9x − 18)/(x+2)
= (4x^2 -9) = (2x +3)(2x-3)

So the other two solutions are
x = 3/2 and x = -3/2

To solve the given equation using factoring by grouping, follow these steps:

Step 1: Group the terms
Rearrange the equation by grouping the terms in pairs:
(4x^3 + 8x^2) + (-9x − 18) = 0

Step 2: Factor out common terms
Now, factor out the greatest common factor from each group:
4x^2(x + 2) - 9(x + 2) = 0

Step 3: Factor out the common binomial
Notice that the binomial (x + 2) is common to both terms. Factor it out:
(x + 2)(4x^2 - 9) = 0

Step 4: Set each factor equal to zero and solve
Now, set each factor equal to zero and solve for x:
x + 2 = 0 ---> x = -2
4x^2 - 9 = 0

Step 5: Solve the quadratic equation
Solve the quadratic equation by factoring or using the quadratic formula:
4x^2 - 9 = 0
(2x + 3)(2x - 3) = 0

Now, set each factor equal to zero and solve for x:
2x + 3 = 0 ---> x = -3/2
2x - 3 = 0 ---> x = 3/2

So, the solutions to the equation 4x^3 + 8x^2 - 9x - 18 = 0 are:

x = -2, x = -3/2, x = 3/2

Among these solutions, -2 is the smallest value, 3/2 is the middle value, and 3/2 is the largest value.