The given problem involves the polynomial $ax^4 - 13x^3 + 27x^2 - 38x + b$, which has $x^2 - 6x + 8$ as a factor. You are asked to find the value of $a + b$.

Solution:

Since $x^2 - 6x + 8$ is a factor, the polynomial can be divided by this quadratic. Let's factor $x^2 - 6x + 8$:

$x^2 - 6x + 8 = (x - 2)(x - 4)$

Now, we express the polynomial $ax^4 - 13x^3 + 27x^2 - 38x + b$ as:

$(ax^4 - 13x^3 + 27x^2 - 38x + b) = (x^2 - 6x + 8)(qx^2 + rx + s)$

Expanding the product:

$(x^2 - 6x + 8)(qx^2 + rx + s) = qx^4 + rx^3 + sx^2 - 6qx^3 - 6rx^2 - 6sx + 8qx^2 + 8rx + 8s$

Now, let's collect terms:

$= qx^4 + (r - 6q)x^3 + (s - 6r + 8q)x^2 + (-6s + 8r)x + 8s$

We compare this with the original polynomial $ax^4 - 13x^3 + 27x^2 - 38x + b$, which gives the following system of equations:

1. $q = a$
2. $r - 6q = -13$
3. $s - 6r + 8q = 27$
4. $-6s + 8r = -38$
5. $8s = b$

Solving the system:

From equation (1): $q = a$

From equation (2): $r - 6a = -13 \quad \Rightarrow \quad r = 6a - 13$

Substitute $r = 6a - 13$ into equation (3):

$s - 6(6a - 13) + 8a = 27$ $s - 36a + 78 + 8a = 27$ $s - 28a + 78 = 27 \quad \Rightarrow \quad s = 28a - 51$

Substitute $s = 28a - 51$ and $r = 6a - 13$ into equation (4):

$-6(28a - 51) + 8(6a - 13) = -38$ $-168a + 306 + 48a - 104 = -38$ $-120a + 202 = -38 \quad \Rightarrow \quad -120a = -240 \quad \Rightarrow \quad a = 2$

Substitute $a = 2$ into the expressions for $r$ and $s$:

$r = 6(2) - 13 = 12 - 13 = -1$ $s = 28(2) - 51 = 56 - 51 = 5$

Now, use equation (5) to find $b$:

$8s = b \quad \Rightarrow \quad 8(5) = b \quad \Rightarrow \quad b = 40$

Finally, $a + b = 2 + 40 = 42$.

Thus, the value of $a + b$ is $\boxed{42}$.

Would you like more details on any of these steps? Here are five additional questions to explore:

1. How do you factor polynomials using synthetic division?
2. What methods can you use to check if a quadratic is a factor of a polynomial?
3. How do you derive the system of equations for polynomial division?
4. What are other ways to solve for coefficients in polynomial factorization?
5. How do you expand and simplify products of polynomials?

Tip: When comparing coefficients in polynomials, always organize terms carefully to avoid errors.