Check a repeated root and include isolated equality solutions

The polynomial has roots −3,−1,1. At −3 the squared factor makes the graph touch the axis without changing sign; the other two roots cross. The sketch is positive on the two intervals left of −1 except at −3, negative between −1 and 1, and positive beyond 1.

Test signs on $(-\infty,-3),(-3,-1),(-1,1),(1,\infty)$: +,+,−,+. For (i), exclude all zeros: x<−3 or −3<x<−1 or x>1. For (ii), include the zeros as well as the negative interval: x=−3 or −1≤x≤1.

For ≥ or ≤, check every critical value carefully. Do not assume signs always alternate: a repeated root with even multiplicity preserves the sign, as at −3. The source contrasts this with Example 1’s simple roots; more generally odd multiplicity changes sign and even multiplicity does not.