【Explanation】:
1. The equation \(x^2 = -31\) has no real solutions since the square of a real number cannot be negative. However, it does have complex solutions, which are ± (i sqrt(31)).
2. For the equation \((x-6)^2-7=0\), by rearranging and taking the square root, we get two solutions: \(x = 6 - \sqrt{7}\) and \(x = 6 + \sqrt{7}\).
3. The equation \(5x^2-1=4\) can be rearranged to \(5x^2 = 5\), which gives \(x^2 = 1\). Taking the square root, we get two solutions: \(x = 1\) and \(x = -1\).
4. For the equation \(36=x^2-4\), rearranging gives \(x^2 = 40\). Taking the square root, we get two solutions: \(x = 2 \sqrt{10}\) and \(x = -2 \sqrt{10}\).
5. The equation \((2x+1)^2=121\) can be simplified to \((2x+1)^2 = 11^2\). Taking the square root, we get two solutions: \(x = 5\) and \(x = -6\).
