2. (a) Express \( x^{2}+8 x-11 \) in the form \( (x+a)^{2}+b \), where \( a \) and \( b \) are constants.

To express \( x^2 + 8x - 11 \) in the form \( (x + a)^2 + b \), we begin by completing the square. Start with the quadratic part \( x^2 + 8x \). We take half of 8, which is 4, and square it, giving us 16. We can write: \[ x^2 + 8x = (x + 4)^2 - 16 \] Now, substitute this back into the original expression: \[ x^2 + 8x - 11 = (x + 4)^2 - 16 - 11 \] Combine the constants: \[ = (x + 4)^2 - 27 \] Thus, we have \( a = 4 \) and \( b = -27 \), so the expression in the required form is: \[ (x + 4)^2 - 27 \]