To express \(4x^2 - 12x + 13\) in the form \((2x + a)^2 + b\), we start by expanding \((2x + a)^2\):

\((2x + a)^2 = (2x)^2 + 2 \cdot 2x \cdot a + a^2 = 4x^2 + 4ax + a^2\).

We want this to match \(4x^2 - 12x + 13\). Comparing coefficients, we have:

\(4ax = -12x\) implies \(4a = -12\), so \(a = -3\).

Substitute \(a = -3\) into \(a^2\):

\(a^2 = (-3)^2 = 9\).

Now, compare the constant terms:

\(a^2 + b = 13\) implies \(9 + b = 13\), so \(b = 4\).

Thus, the expression is \((2x - 3)^2 + 4\).

To express \(x^2 - 4x + 8\) in the form \((x-a)^2 + b\), we complete the square:

1. Start with the expression: \(x^2 - 4x + 8\).

2. Take the coefficient of \(x\), which is \(-4\), halve it to get \(-2\), and square it to get \(4\).

3. Add and subtract \(4\) inside the expression:

\(x^2 - 4x + 4 - 4 + 8\)

4. Rewrite as a perfect square and simplify:

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

To express \(3x^2 - 12x + 7\) in the form \(a(x + b)^2 + c\), follow these steps:

1. Factor out the coefficient of \(x^2\) from the quadratic and linear terms:

\(3(x^2 - 4x) + 7\)

2. Complete the square for the expression inside the parentheses:

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

3. Substitute back into the expression:

\(3((x - 2)^2 - 4) + 7\)

4. Distribute the 3 and simplify:

\(3(x - 2)^2 - 12 + 7\)

5. Combine the constants:

\(3(x - 2)^2 - 5\)

Thus, \(a = 3\), \(b = -2\), and \(c = -5\).

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

1. Start with the expression: \(x^2 + 6x + 2\).

2. Take half of the coefficient of \(x\), which is 6, giving 3, and square it: \(3^2 = 9\).

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

4. Rewrite as a perfect square: \((x + 3)^2 - 9 + 2\).

5. Simplify the constants: \((x + 3)^2 - 7\).

Thus, \(a = 3\) and \(b = -7\).

Given the function \(f(x) = x^2 + ax + b\) with roots \(x = 1\) and \(x = 9\), we can express it as:

\(f(x) = (x - 1)(x - 9)\)

Expanding this, we get:

\(f(x) = x^2 - 10x + 9\)

Thus, \(a = -10\) and \(b = 9\).

To find the vertex, use the vertex formula \(x = \frac{-b}{2a}\) or symmetry:

\(x = \frac{1 + 9}{2} = 5\)

Substitute \(x = 5\) into \(f(x)\):

\(f(5) = 5^2 - 10 \times 5 + 9 = -16\)

Thus, the vertex is \((5, -16)\).