To determine the number in the left-hand box for synthetic division when dividing [tex]\( x^3 + 12x^2 + 20x - 96 \)[/tex] by [tex]\( x - 2 \)[/tex], follow these steps:
1. Identify the root of the divisor: The divisor given is [tex]\( x - 2 \)[/tex]. To find the root, set [tex]\( x - 2 = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 2 = 0 \implies x = 2
\][/tex]
2. Use the root in synthetic division: The root [tex]\( x = 2 \)[/tex] is the number that goes in the left-hand box of the synthetic division setup.
Thus, the number in the left-hand box for the synthetic division of [tex]\( x^3 + 12x^2 + 20x - 96 \)[/tex] by [tex]\( x - 2 \)[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
