Further, we should all remember to draw a graph as our first step. Symbolab, Desmos and Geogebra (my favourite) are all fantastic graphing software.
Also, for a quadratic if you know the zeroes (roots), it's half-way between them. So for the equation -(x-1)(x-5) = 0, the roots are x=1 and x=5, and the mid-point (maximum) is at x=3. Your graphing will confirm this.
Your stepdaughter might also have learnt that the mid-point of a quadratic ax^2 + bx + c = 0 is x=-b/2a. (This can be derived by setting the derivative equal to zero, but is generally just given to the students.) Her teacher may be expecting her to use that formula.
Alternatively the students might be required to determine and plot various data points using, say, Excel, and find the maximum this way.
The other point is that solutions often do not spring to mind immediately. That is why I ask students to send me their problems before our tutoring sessions. I often need time to think about them.
The fun of math is this exploration to find an answer. So try to get problems and give yourselves time (days) to explore them together.
a(x - b)^2 + c
In lucky cases, fourth-order polynomials could be put in such a form. Odd-degree polynomials will not have a unique maximum.
I recently-ish helped my stepdaughter with an algebra assignment where she was asked to find the maximum of some polynomial. I totally blanked on how to do this with only the tools that she was supposed to have. I didn't know how to do it or explain it without calculus.