To write a quadratic function with roots 37 and $P$, we use the fact that a quadratic equation with roots $x_1$ and $x_2$ can be written in factored form as:

$f(x) = (x - x_1)(x - x_2)$

For your problem, the roots are $37$ and $P$, so the quadratic function is:

$f(x) = (x - 37)(x - P)$

Since you want the leading coefficient to be 1, there's no need to multiply by any constant. Expanding this:

$f(x) = x^2 - Px - 37x + 37P$

$f(x) = x^2 - (P + 37)x + 37P$

Thus, the quadratic function is:

$f(x) = x^2 - (P + 37)x + 37P$

Would you like further details or explanations on this process?

Here are five related questions to expand on this:

1. How would the function change if the leading coefficient were not 1?
2. What are the implications of the sum and product of the roots in a quadratic equation?
3. How can you verify if a quadratic equation has certain roots?
4. What does the discriminant tell you about the nature of the roots?
5. How would this quadratic function change if both roots were negative?

Tip: The sum of the roots of a quadratic function $ax^2 + bx + c = 0$ is given by $-b/a$, and the product is $c/a$.