Final answer:
The factored form of the inequality x³+6x²+9x<0 is x(x+3)²<0. The solution to the inequality are the intervals where the expression is negative, which is (-∞,-3)∪(-3,0).
Step-by-step explanation:
To find all real values of x such that x³+6x²+9x<0, first factor the left side of the inequality. You can factor out an x, getting x(x²+6x+9)<0. The quadratic x²+6x+9 can be factored further as (x+3)². Therefore, the factored inequality is x(x+3)²<0. The zeros of the expression are x=0 and x=-3 (with a multiplicity of 2).
The inequality is satisfied between the roots where the expression is negative. Since (x+3)² is always non-negative, the sign is determined by the x. Thus, the solution set is the interval when x is less than zero but not including the point where x+3 equals zero since squaring it would result in a non-negative number. Therefore, the real values of x that satisfy the inequality are in the interval (-∞,-3)∪(-3,0).