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Finding the Zeroes of Functions

$Let\ h(x)=x^{3}+6x^{2}+5x.\ Find\ all\ the\ zeroes\ of\ the\ function.$

$x=\ ?\ when\ y=0$

$0=x^{3}+6x^{2}+5x$

Step 1: factor out an “x”

$0=x(x^{2}+6x+5)$

Step 2: factor $x^{2}+6x+5$

$(x+m)(x+n)=x^{2}+mx+nx+mn$

$m+n=6$

$mn=5$

$factor\ of\ 5:1,\ 5\ \ \ m=1,\ n=5$

$(x+1)(x+5)$

Step 3: combine & solve

$0=x(x+1)(x+5)$

$x=0,\ x+1=0,\ x+5=0$

$x+1-1=0-1\rightarrow x=-1$

$x+5-5=0-5\rightarrow x=-5$

$zeroes:\ 0,\ -1,\ -5$