B1. Solve the equation: $4 x^{4}+2 x^{3}-4 x^{2}-x=-1$. Calculate the difference of the squares of the sum of the rational solutions and the sum of the irrational solutions of the equation.
B1. We can solve the equation using Horner's algorithm and obtain the solutions $x_{1}=$ $-1, x_{2}=\frac{1}{2}, x_{3}=\frac{\sqrt{2}}{2}, x_{4}=-\frac{\sqrt{2}}{2}$. We determine that the first two solutions are rational, and the other two are irrational. We calculate the square of the sum of the first two solutions and get $\frac{1}{4}$, and the square of the sum of the other two solutions and get 0. Finally, we calculate the difference of the squares of the sum of the rational solutions and the sum of the irrational solutions of the equation and get the solution $\frac{1}{4}$.

Using the appropriate method to solve the equation ........................................................................................

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Calculating the irrational solutions of the equation $x_{3}=\frac{\sqrt{2}}{2}, x_{4}=-\frac{\sqrt{2}}{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . .2$ points

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![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-24.jpg?height=52&width=1725&top_left_y=1910&top_left_x=177)

Calculating the difference of the squares of the sum of the rational solutions and the sum of the irrational solutions of the equation $\frac{1}{4} \ldots 1$ point