Find the remainder when $3x^3 - x^2 + 4x + 2$ is divided by $x^2 + 1$. The remainder when $x^3 + ax$, where $a$ is a constant, is divided by $x + 2$ is the same as the remainder when it is divided by $x - 1$. Find the value of $a$.

1. Perform polynomial long division to find the $x$ term of the quotient

$\begin{array}{r|l} x^2 + 1 & 3x^3 - x^2 + 4x + 2 \\ & 3x^3 \quad\quad + 3x \\ \hline & \quad -x^2 + x + 2 \end{array}$

2. Find the constant term of the quotient and the final remainder

$\begin{array}{r|l} x^2 + 1 & 3x^3 - x^2 + 4x + 2 \\ & 3x^3 \quad\quad + 3x \\ \hline & \quad -x^2 + x + 2 \\ & \quad -x^2 \quad\quad - 1 \\ \hline & \quad\quad\quad\quad x + 3 \end{array}$
The remainder is $x + 3$.

1. Find the expression for the remainder when divided by $x + 2$.

Let $P(x) = x^3 + ax$.
By the Remainder Theorem, the remainder when $P(x)$ is divided by $x + 2$ is $P(-2)$.
$P(-2) = (-2)^3 + a(-2) = -8 - 2a$

2. Find the expression for the remainder when divided by $x - 1$.

The remainder when $P(x)$ is divided by $x - 1$ is $P(1)$.
$P(1) = (1)^3 + a(1) = 1 + a$

3. Equate the two remainders and solve for $a$.

Since the remainders are the same:
$-8 - 2a = 1 + a$
$3a = -9$
$a = -3$