Question

y=(x+1)^2 (x-2)^3 (x+3) solve the inequality y>0

Answer 1

are all those being multiplied together?

Answer 2

Yes

Answer 3

ignore the first term, because it is always positive

Answer 4

so consider only
\[(x-2)^3(x+3)>0\] and then note that \((x-2)^3\) is greater than zero if and only if \(x-2>0\) so now consider only
\[(x-2)(x+3)>0\]

Answer 5

\((x-2)(x+3)\) is a parabola that opens up, so it is positive outside the zeros and negative between them
the zeros are \(-3\) and \(2\)
so it is negative on \((-3,2)\) and positive on
\[(-\infty,-3)\cup (2,\infty)\]

Answer 6

oh actually if it is \(y>0\) you have to exclude \(-1\) as well, because at \(x=-1\) it is zero also

Answer 7

so maybe best to write
\(x<-3\) or \(x>2\) and \(x\neq -1\)