Write as a series in ascending powers of x, as far as the term in x3 , and state the values of x for which the expansion is valid: root(4+x).
I don't know what this means or how to do it.
Use the binomial expansion, which, for (a + x)n = an + nan-1x + n(n-1)an-2x2/2! + n(n-1)(n-2)an-3x3/3! + ...
Here: a = 4, n = 1/2 (because √(4 + x) = (4 + x)1/2), and you don't need to go further than the terms written above because these go as far as the x3 term.