\( a ^ { 4 } + a ^ { 4 } \beta ^ { 4 } - 4 a ^ { 2 } \beta ^ { 2 } + \beta ^ { 4 } = 4 a \beta \)

To solve the equation \[ a^4 + a^4 \beta^4 - 4a^2 \beta^2 + \beta^4 = 4a\beta, \] let's first rearrange it to bring all terms to one side: \[ a^4 + a^4 \beta^4 - 4a^2 \beta^2 + \beta^4 - 4a\beta = 0. \] Next, we can group the terms: \[ a^4(1 + \beta^4) + \beta^4 - 4a^2 \beta^2 - 4a\beta = 0. \] This is a polynomial in terms of \(a\) and \(\beta\). To analyze it further, we can try substituting specific values for \(a\) and \(\beta\) or factor the polynomial if possible. Let's try substituting \(a = 0\): \[ 0 + 0 - 0 + \beta^4 = 0 \implies \beta^4 = 0 \implies \beta = 0. \] So, one solution is \( (a, \beta) = (0, 0) \). Next, let's try \( \beta = 0 \): \[ a^4 + 0 - 0 - 0 = 0 \implies a^4 = 0 \implies a = 0. \] This confirms that \( (0, 0) \) is indeed a solution. Now, let's check for other potential solutions. We can rewrite the equation in a more manageable form. Let's isolate the terms involving \(a\): \[ a^4(1 + \beta^4) - 4a^2 \beta^2 + \beta^4 - 4a\beta = 0. \] This is a quartic equation in \(a\). To find other solutions, we can use numerical methods or graphing techniques, or we can analyze the discriminant of the polynomial. However, without loss of generality, we can also try specific values for \(\beta\) to see if we can find rational solutions. For example, if we set \(\beta = 1\): \[ a^4 + a^4 - 4a^2 + 1 - 4a = 0 \implies 2a^4 - 4a^2 - 4a + 1 = 0. \] This is a quartic equation in \(a\). Solving this analytically or numerically can yield additional solutions. In summary, we have found one solution \( (0, 0) \). Further exploration with specific values or numerical methods may yield additional solutions.