Please help me to answer this, thank you!
Find the equation of the curve that satisfies the differential equation f"(x) = 6x - 12 and which passes through the points (0, -8) and (1, -1).
I do not know if you understood what topsquak said, but when you have,
$\displaystyle f''(x)=6x-12$
It means if you integrate both sides you have,
$\displaystyle f'(x)=3x^2-12x+a$ with some constant $\displaystyle a$,
the reason to why f''(x) becomes f'(x) is because when you integrate a derivative of a function you get back the function (definition of integral). So when you have f''(x) integrating once would yield f'(x) so you need one more time:
$\displaystyle f(x)=x^3-6x^2+ax+b$
Then use topsqaurk post to complete the problem.