The given problem involves completing the square for the quadratic function:

$h(x) = x^2 + 3x - 18$

The equation provided shows the function in vertex form, where we need to complete the square:

$h(x) = \left( x + 1.5 \right)^2 + \square$

Step-by-step solution:

1. Start with the expression $x^2 + 3x$.
2. The coefficient of $x$ is 3. Divide this by 2: $\frac{3}{2} = 1.5$, and then square it: $(1.5)^2 = 2.25$.
3. To complete the square, add and subtract 2.25 inside the equation:

$h(x) = (x^2 + 3x + 2.25) - 2.25 - 18$

4. Now factor the quadratic expression:

$h(x) = (x + 1.5)^2 - 2.25 - 18$

5. Simplify the constants:

$h(x) = (x + 1.5)^2 - 20.25$

Thus, the function in completed square form is:

$h(x) = (x + 1.5)^2 - 20.25$

So the missing values are:

$h(x) = 1(x + 1.5)^2 - 20.25$

Would you like more details on completing the square, or do you have any further questions?

Related Questions:

1. How can completing the square help in finding the vertex of a quadratic function?
2. What are other methods to convert a quadratic equation into vertex form?
3. How does completing the square relate to solving quadratic equations?
4. Can you always complete the square for any quadratic function? Why or why not?
5. How is completing the square used in the quadratic formula derivation?

Tip: Completing the square is a powerful tool not only to rewrite quadratics but also to solve them and understand their geometric properties (like finding the vertex).