(a) To express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\), we complete the square:

Start with \(x^2 + 6x\). Take half of the coefficient of \(x\), which is 3, and square it to get 9.

Add and subtract 9 inside the expression: \(x^2 + 6x = (x^2 + 6x + 9) - 9 = (x + 3)^2 - 9\).

Now, include the constant term 5: \((x + 3)^2 - 9 + 5 = (x + 3)^2 - 4\).

Thus, \(x^2 + 6x + 5 = (x + 3)^2 - 4\).

(b) The transformation from \(y = x^2\) to \(y = x^2 + 6x + 5\) involves completing the square as shown in part (a): \(y = (x + 3)^2 - 4\).

This represents a translation of the graph of \(y = x^2\) by \(\begin{pmatrix} -3 \\ -4 \end{pmatrix}\), which means 3 units to the left and 4 units down.