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

Step-by-step solution:

1. Factor of the Polynomial: The polynomial is divisible by $x^2 - 6x + 8$, which can be factored further as: $x^2 - 6x + 8 = (x - 2)(x - 4)$ This tells us that $x = 2$ and $x = 4$ are roots of the polynomial.

2. Using Root Substitution: Since $x = 2$ is a root, substitute $x = 2$ into the polynomial $ax^4 - 13x^3 + 27x^2 - 38x + b$ and set it equal to 0: $a(2)^4 - 13(2)^3 + 27(2)^2 - 38(2) + b = 0$ Simplifying this: $16a - 13(8) + 27(4) - 38(2) + b = 0$ $16a - 104 + 108 - 76 + b = 0$ $16a + b - 72 = 0$ Therefore, we have the first equation: $16a + b = 72 \quad \text{(Equation 1)}$

3. Using the Second Root: Now, substitute $x = 4$ into the polynomial: $a(4)^4 - 13(4)^3 + 27(4)^2 - 38(4) + b = 0$ Simplifying this: $256a - 13(64) + 27(16) - 38(4) + b = 0$ $256a - 832 + 432 - 152 + b = 0$ $256a + b - 552 = 0$ Therefore, we have the second equation: $256a + b = 552 \quad \text{(Equation 2)}$

4. Solving the System of Equations: We now have two equations:

   1. $16a + b = 72$
   2. $256a + b = 552$

   Subtract Equation 1 from Equation 2: $(256a + b) - (16a + b) = 552 - 72$ Simplifying: $240a = 480$ $a = 2$

5. Finding $b$: Substitute $a = 2$ into Equation 1: $16(2) + b = 72$ $32 + b = 72$ $b = 40$

6. Calculating $a + b$: $a + b = 2 + 40 = 42$

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

Would you like more details or have any questions about this solution? Here are 5 related questions you could explore:

1. How do we check if a polynomial is divisible by another polynomial?
2. What happens if a polynomial has more than one factor, such as quadratic and linear factors?
3. Can a polynomial have a root at $x = 0$ without an explicit constant term?
4. How does synthetic division work for factoring higher-degree polynomials?
5. What are the methods for solving systems of equations involving polynomials?

Tip: Always double-check your factorization of polynomials to ensure that you are working with the correct roots and factors!