What do they mean by 'for all x' in (a)?
I think red_dog did an excellent job on this question. However, i shall take baby steps and leave out the formalism, hoefully you can get the concept from this.
We have
we want this to be true no matter what we choose. we will do the classic move of equating coefficients to accomplish this. Let's expand the right side, we get:
Now group like powers of
so now we match up the coefficients. since the two sides are equal, we must have the same number of each power of on both sides.
we have 1 on each side, so we're good there
on the left we have -8x on the right we have -2px, they must be equal, so we must have:
on the left we have 0 as the constant term, on the right we have as the constant term. since we want the constants to be the same as well for the sake of equality, we must have:
..........since
so those are the two values we need. so no matter what is, those values for and will work
did you get it?