1. State the problem. Determine the x-intercept(s) of the function $y=3(x+6)^4-48$. 2. Set the function equal to zero to find x-intercepts. Solve $0=3(x+6)^4-48$. 3. Isolate the power term. Add 48 to both sides to get $3(x+6)^4=48$. Divide both sides by 3 to get $(x+6)^4=16$. 4. Solve the fourth-power equation. Since $a^4=16$ implies $|a|=2$, we set $a=x+6$ and get $|x+6|=2$. Therefore $x+6=2$ or $x+6=-2$. Solving gives $x=-4$ or $x=-8$. 5. Verify by factoring (optional). Rewrite $(x+6)^4-16$ as $((x+6)^2-4)((x+6)^2+4)$. The factor $((x+6)^2+4)$ has no real roots, while $((x+6)^2-4)=(x+8)(x+4)$ gives $x=-8$ and $x=-4$. 6. Final answer. The x-intercepts are $(-8,0)$ and $(-4,0)$.