To solve the equation
\[ \sqrt[5]{3x^2 - 10} + 7 = 4, \]
first, we isolate the radical term:
\[ \sqrt[5]{3x^2 - 10} = 4 - 7, \]
which simplifies to
\[ \sqrt[5]{3x^2 - 10} = -3. \]
Next, we eliminate the fifth root by raising both sides to the power of 5:
\[ 3x^2 - 10 = (-3)^5. \]
Calculating \((-3)^5\):
\[ (-3)^5 = -243. \]
Now we have:
\[ 3x^2 - 10 = -243. \]
To isolate \(3x^2\), we add 10 to both sides:
\[ 3x^2 = -243 + 10, \]
which simplifies to:
\[ 3x^2 = -233. \]
Next, divide both sides by 3:
\[ x^2 = \frac{-233}{3}. \]
Since \(x^2\) cannot be negative, this implies that there are no real solutions to the equation.
Thus, the final conclusion is that there are no solutions for the equation.
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