To express \(4x^2 - 24x + p\) in the form \(a(x + b)^2 + c\), follow these steps:

1. Start with the expression: \(4x^2 - 24x + p\).

2. Factor out the 4 from the first two terms: \(4(x^2 - 6x) + p\).

3. Complete the square for \(x^2 - 6x\):

- Take half of the coefficient of \(x\), which is -6, to get -3.

- Square it to get 9.

4. Rewrite \(x^2 - 6x\) as \((x - 3)^2 - 9\).

5. Substitute back: \(4((x - 3)^2 - 9) + p\).

6. Expand and simplify: \(4(x - 3)^2 - 36 + p\).

7. Therefore, \(a = 4\), \(b = -3\), and \(c = p - 36\).