Answer
zero: -4
P(x)=(x+4)^2
Work Step by Step
P(x)=x^{3}+12x^{2}+48x+64
P(-x)=(-x)^{3}+12(-x)^{2}+48(-x)+64
a_{0}=64 p:+/- 1,+/-2,+/-4,+/-8,+/-16,
+/-64
a_{n}= q:+/-1
test x=-1 by synthetic division
-1| 1 12 48 64
-1 -13 -35
-- -- --
-1 13 35 129 =>P(-1)\ne 0
-1 is not a zero
-4| 1 12 48 64
-4 -32 -64
-- -- --
1 8 16 | 0 => P(-4)=0
x=-4 is a zero, (x+4) is a factor of P(x).
The bottom row gives coefficient of the quotient
P(x)=(x+4)(x^{2}+8x+16
=(x+4)(x+4)=(x+4)^{2}
so, the zero is -4
