To solve the equation 2^x + 2 * 3^x + 1.5^x = 10800, we can start by simplifying the equation.

Let's rewrite 1.5^x as (3/2)^x.

2^x + 2 * 3^x + (3/2)^x = 10800

Now, let's simplify further by multiplying through by 2^x to eliminate the fractions:

(2^x) * (2^x) + 2 * (2^x) * (3^x) + (2^x) * ((3/2)^x) = 10800 * (2^x)

2^(2x) + 2 * 2^x * 3^x + 2^x * (3/2)^x = 10800 * 2^x

Now, let's substitute y = 2^x:

y^2 + 2 * y * 3^x + y * (3/2)^x = 10800 * y

y^2 + 2 * y * 3^x + y * (3/2)^x - 10800 * y = 0

Now, we have a quadratic equation in terms of y. We can solve this equation using the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 2 * 3^x, and c = (3/2)^x - 10800.

Substituting these values into the quadratic formula, we get:

y = (-2 * 3^x ± √((2 * 3^x)^2 - 4 * (3/2)^x * (3/2)^x - 4 * (3/2)^x * 10800)) / (2 * 1)

Simplifying further:

y = (-2 * 3^x ± √(4 * 9^x - 4 * (9/4)^x - 4 * (3/2)^x * 10800)) / 2

y = (-2 * 3^x ± √(4 * 9^x - 4 * (9/4)^x - 4 * 3^x * 10800)) / 2

Now, we can solve for y and then substitute back to solve for x. However, this equation does not have a simple algebraic solution. We can use numerical methods, such as graphing or iterative methods, to approximate the value of x that satisfies the equation.