$(-7,\infty)$ See the figure below.
Work Step by Step
Distribute $-2$ and then add $6$ to both sdies to obtain $$ - 2(x + 3 )\lt 8 \Longrightarrow -2x-6 \lt 8\\ \Longrightarrow -2x-6 +6\lt 8+6 \Longrightarrow -2x\lt 14 .$$ Divide by $-2$ and reverse the inquality since we divide by a negative value, we get $$ -2x\lt 14 \Longrightarrow x \gt -7 .$$ In terms of intervals, $x\in (-7,\infty)$. To graph, plot a circle at $x=-7$ then shade the region to its right. See the figure below.