To solve this, let's follow the steps to convert the quadratic function $f(x) = 18x + 3x^2$ into vertex form:

Step 1: Write the function in standard form

The given function is already in standard form:

$f(x) = 3x^2 + 18x$

Step 2: Factor out the coefficient of $x^2$ from the first two terms

Factor out 3 from the terms involving $x$:

$f(x) = 3(x^2 + 6x)$

Step 3: Form a perfect square trinomial

To complete the square, take half of the coefficient of $x$ (which is 6), square it, and add it inside the parentheses:

$\text{Half of } 6 \text{ is } 3. \quad 3^2 = 9$

So, add and subtract 9 inside the parentheses:

$f(x) = 3(x^2 + 6x + 9) - 3(9)$

Step 4: Write the trinomial as a binomial squared

The trinomial $x^2 + 6x + 9$ can be factored as $(x + 3)^2$:

$f(x) = 3(x + 3)^2 - 27$

Conclusion:

The missing value in the binomial is 3.

So, the correct answer is:

$\textbf{3}$

Do you want more details or have any questions?

Here are 5 related questions you might consider:

1. How do you find the vertex of a quadratic function in vertex form?
2. What is the significance of completing the square in solving quadratic equations?
3. How do you convert a quadratic function from vertex form back to standard form?
4. What are the steps to factor a quadratic equation?
5. How does the discriminant help in solving quadratic equations?

Tip: When completing the square, always remember to balance the equation by adding and subtracting the same value inside the parentheses.