The equation in the image shows:

$(x - 4) \cdot P(x) = x^3 - 2x^2 + ax + b$

We are asked to determine the value of $b$.

To solve this, let's express $P(x)$ as a polynomial and use polynomial division or equating coefficients. Let $P(x)$ be a quadratic polynomial:

$P(x) = x^2 + px + q$

Now, expand $(x - 4) \cdot P(x)$:

$(x - 4)(x^2 + px + q) = x^3 + px^2 + qx - 4x^2 - 4px - 4q$

Simplify the expression:

$x^3 + (p - 4)x^2 + (q - 4p)x - 4q$

Now, compare this with the original equation:

$x^3 - 2x^2 + ax + b$

By equating the coefficients of the corresponding powers of $x$, we get the following system of equations:

1. Coefficient of $x^2$: $p - 4 = -2$ → $p = 2$
2. Coefficient of $x^1$: $q - 4p = a$ → $q - 8 = a$
3. Constant term: $-4q = b$

Now, we know that $p = 2$, so substitute into the third equation for $b$:

$-4q = b$

Thus, $b = -4q$.

If you would like further clarification or details on the next steps, feel free to ask!

Here are some related questions to consider:

1. What happens if we try a different degree for $P(x)$?
2. How does the value of $a$ affect the function in this equation?
3. Could there be multiple values for $b$ in similar polynomial equations?
4. How can polynomial long division help in these types of problems?
5. What if the given equation had no constant term—how would that affect $P(x)$?

Tip: Always align the degrees of terms when expanding polynomial expressions for easier comparison.