To express \(x^2 - 4x + 5\) in the form \((x + a)^2 + b\), we complete the square:

Start with the expression:

\(x^2 - 4x + 5\)

Take half of the coefficient of \(x\), square it, and add and subtract it inside the expression:

\(x^2 - 4x + 4 - 4 + 5\)

Rewrite as a perfect square:

\((x - 2)^2 + 1\)

Thus, the expression in the form \((x + a)^2 + b\) is \((x - 2)^2 + 1\).

The minimum point occurs at \(x = 2\), giving the coordinates \((2, 1)\).

To express \(4x^2 - 24x + p\) in the form \(a(x + b)^2 + c\), follow these steps:

1. Start with the expression: \(4x^2 - 24x + p\).

2. Factor out the 4 from the first two terms: \(4(x^2 - 6x) + p\).

3. Complete the square for \(x^2 - 6x\):

- Take half of the coefficient of \(x\), which is -6, to get -3.

- Square it to get 9.

4. Rewrite \(x^2 - 6x\) as \((x - 3)^2 - 9\).

5. Substitute back: \(4((x - 3)^2 - 9) + p\).

6. Expand and simplify: \(4(x - 3)^2 - 36 + p\).

7. Therefore, \(a = 4\), \(b = -3\), and \(c = p - 36\).

To express \(2x^2 - 4x + 1\) in the form \(a(x + b)^2 + c\), we complete the square:

Start with the quadratic expression:

\(2x^2 - 4x + 1\)

Factor out the 2 from the first two terms:

\(2(x^2 - 2x) + 1\)

Complete the square inside the parentheses:

\(x^2 - 2x = (x - 1)^2 - 1\)

Substitute back:

\(2((x - 1)^2 - 1) + 1\)

Distribute the 2:

\(2(x - 1)^2 - 2 + 1\)

Simplify:

\(2(x - 1)^2 - 1\)

Thus, \(a = 2\), \(b = -1\), \(c = -1\).

The minimum point \(A\) is at \((1, -1)\).

(a) To express \(y = 4x^2 + 20x + 6\) in the form \(y = a(x + b)^2 + c\), complete the square:

Factor out 4 from the quadratic terms: \(y = 4(x^2 + 5x) + 6\).

Complete the square for \(x^2 + 5x\):

\(x^2 + 5x = (x + \frac{5}{2})^2 - (\frac{5}{2})^2\).

Substitute back: \(y = 4((x + \frac{5}{2})^2 - \frac{25}{4}) + 6\).

Simplify: \(y = 4(x + \frac{5}{2})^2 - 25 + 6\).

\(y = 4(x + \frac{5}{2})^2 - 19\).

(b) Solve \(4(x + \frac{5}{2})^2 - 19 = 45\):

\(4(x + \frac{5}{2})^2 = 64\).

\((x + \frac{5}{2})^2 = 16\).

\(x + \frac{5}{2} = \pm 4\).

\(x = -\frac{5}{2} + 4 = \frac{3}{2}\) or \(x = -\frac{5}{2} - 4 = -\frac{13}{2}\).

(c) The stationary point is at \(x = -\frac{5}{2}\), giving \(y = -19\).

Stationary point: \((-\frac{5}{2}, -19)\).

(a) To express \(x^2 - 8x + 11\) in the form \((x + p)^2 + q\), complete the square:

Start with \(x^2 - 8x\).

Take half of the coefficient of \(x\), square it, and add and subtract it inside the expression:

\(x^2 - 8x = (x - 4)^2 - 16\).

Now add 11 to balance the equation:

\((x - 4)^2 - 16 + 11 = (x - 4)^2 - 5\).

Thus, \(p = -4\) and \(q = -5\).

(b) Solve \((x - 4)^2 - 5 = 1\).

First, add 5 to both sides:

\((x - 4)^2 = 6\).

Take the square root of both sides:

\(x - 4 = \pm \sqrt{6}\).

Finally, solve for \(x\):

\(x = 4 \pm \sqrt{6}\).