Hi there. Can "x^2 - 10x + 22 = -2" be solved by taking the square root? Can "4x^2 + 8x + 3 = 0" be solved by taking the square root? If so, please show me an explanation.

You probably mean, "completing the square", where we take the square root near the end of the process

x^2 - 10x + 22 = -2
x^2 - 10x + ..... = -24 + .....
x^2 - 10x + 25 = -24 + 25
(x-5)^2 = 1
take √ of both sides
x - 5 = ± √1 = ± 1

x = 5 ± 1
x = 6 or x = 4