[User]
x^5 + 10x^3 + 20x - 4 = 0. Find a closed form expression for the real roots of x in the simplest form.

Present the answer in LaTex format: \boxed{Your answer}

dataset: MATH


[User]
x^5 + 5x^3 + 5x - 2 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 10x^3 + 20x + 28 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 15x^3 + 45x - 26 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 20x^3 + 80x - 66 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 30x^3 + 180x - 122 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 - 10x^3 + 20x + 36 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 5x^3 + 5x - 30 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 40x^3 + 320x - 258 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 15x^3 + 45x + 28 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 - 5x^3 + 5x - 6 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 25x^3 + 125x - 126 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 10x^3 + 20x + 56 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 5x^3 + 5x + 2 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 30x^3 + 180x - 344 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 - 20x^3 + 80x + 66 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 5x^3 + 5x - 122 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}


[User]
x^5 + 50x^3 + 500x - 1252 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\sqrt{c})^{\frac{1}{5}} + d(a + b\sqrt{c})^{-\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.

Present the answer in LaTex format: \boxed{Your answer}