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Since it's a parabola opening downward, and is shifted up 5 units, you know that it will have two x-intercepts. So, what went wrong?
You lost a "-" sign after the 2nd line.
-5 = -6(x+7)^4
5/6 = (x+7)^4
±(5/6)^(1/4) = x+7
x = -7 ±(5/6)^(1/4) = -7 ± .955