Final answer:
The question appears to relate to a polynomial function and the dimensions of a box, but without additional context or a specific query, it is challenging to provide a precise answer. The student may be seeking help with factoring the polynomial or analyzing the function with calculus, but further clarification is needed to accurately solve the problem.
Step-by-step explanation:
The student has provided a polynomial function f(x) = x^3 + 7x^2 + 4x - 12 and dimensions of a box with length (x + 6) and width (x + 2).
However, the question seems incomplete as there is no clear indication of what needs to be found or solved regarding the polynomial and the dimensions of the box. One can speculate that the student might be seeking to factor the polynomial using the dimensions given, which implies looking for factors of the polynomial that match the given length and width.
Alternatively, the question might be related to finding the volume of the box, but that would require a factorization of the polynomial to match the length and width, or it could be a question about integration or graph analysis indicated by the reference to Figure 7.8 and the various integration notations.
Without further context or a specific question, it is challenging to provide a precise answer. If the student is looking to factor the polynomial in the form of the box dimensions, we would attempt to rewrite f(x) as (x + 6)(x + 2)(Ax + B) for some constants A and B, and find those values through polynomial division or other factoring techniques.
If the question pertains to calculus and the area under the curve represented by f(x), integration would be used between limits x1 and x2 to find the area. More clarification from the student is needed to provide a step-by-step solution.