The given problem is:

When $ax^3 - x^2 + 2x + b$ is divided by $x - 1$, the remainder is 10. When it is divided by $x - 2$, the remainder is 51. We are asked to find the values of $a$ and $b$.

Approach:

By the remainder theorem, if a polynomial $f(x)$ is divided by $x - c$, the remainder is $f(c)$. Applying this theorem to the problem:

1. For division by $x - 1$: $f(1) = 10$ Substituting into the polynomial $f(x) = ax^3 - x^2 + 2x + b$: $f(1) = a(1)^3 - (1)^2 + 2(1) + b = a - 1 + 2 + b = a + b + 1$ So, $a + b + 1 = 10$, which simplifies to: $a + b = 9 \quad \text{(Equation 1)}$

2. For division by $x - 2$: $f(2) = 51$ Substituting into the polynomial: $f(2) = a(2)^3 - (2)^2 + 2(2) + b = 8a - 4 + 4 + b = 8a + b$ So, $8a + b = 51 \quad \text{(Equation 2)}$.

Solving the system of equations:

We now have the system:

1. $a + b = 9$
2. $8a + b = 51$

From the first equation, solve for $b$: $b = 9 - a$

Substitute this into the second equation: $8a + (9 - a) = 51$ $8a - a + 9 = 51$ $7a + 9 = 51$ $7a = 42$ $a = 6$

Substitute $a = 6$ back into the first equation: $6 + b = 9$ $b = 3$