The problem shown in the image asks for division of the polynomial $x^3 + 4x^2 - 3x - 2$ by $x - 1$ using long division. The goal is to find the quotient, $q(x)$, and the remainder, $r(x)$.

Steps for Polynomial Long Division:

1. Divide the first term: Divide the leading term of the dividend $x^3$ by the leading term of the divisor $x$, which gives $x^2$. This is the first term of the quotient, so: $q(x) = x^2$

2. Multiply: Multiply $x^2$ by $x - 1$ to get $x^3 - x^2$.

3. Subtract: Subtract $x^3 - x^2$ from $x^3 + 4x^2 - 3x - 2$, leaving: $(x^3 + 4x^2 - 3x - 2) - (x^3 - x^2) = 5x^2 - 3x - 2$

4. Repeat the process: Divide $5x^2$ by $x$, which gives $5x$. Multiply $5x$ by $x - 1$ to get $5x^2 - 5x$, and subtract: $(5x^2 - 3x - 2) - (5x^2 - 5x) = 2x - 2$

5. Final step: Divide $2x$ by $x$, which gives $2$. Multiply $2$ by $x - 1$ to get $2x - 2$, and subtract: $(2x - 2) - (2x - 2) = 0$

Thus, the quotient is $q(x) = x^2 + 5x + 2$, and there is no remainder, $r(x) = 0$.

Final Answer:

$\frac{x^3 + 4x^2 - 3x - 2}{x - 1} = x^2 + 5x + 2$

Would you like further details or explanations?

Here are 5 related questions:

1. How would you use synthetic division for this same problem?
2. What is the Remainder Theorem, and how could it apply here?
3. How can you verify your result using polynomial multiplication?
4. What happens when you divide by a binomial that is not of the form $x - a$?
5. How does the degree of the divisor impact the long division process?

Tip: Always remember to check your quotient by multiplying it with the divisor to ensure the remainder matches the original polynomial.