The problem asks us to find the remainder when the polynomial $x^4 + 2x^2 - 3x + 7$ is divided by $x + 2$.

1. Problem Description

The problem asks us to find the remainder when the polynomial $x^4 + 2x^2 - 3x + 7$ is divided by $x + 2$.

2. Solution Steps

We can use the Remainder Theorem to find the remainder. The Remainder Theorem states that if we divide a polynomial $f(x)$ by $x - c$, the remainder is $f(c)$. In this case, we are dividing by $x + 2$, so $x - c = x + 2$, which means $c = -2$.

We need to evaluate the polynomial $f(x) = x^4 + 2x^2 - 3x + 7$ at $x = -2$.

$f(-2) = (-2)^4 + 2(-2)^2 - 3(-2) + 7$

$f(-2) = 16 + 2(4) + 6 + 7$

$f(-2) = 16 + 8 + 6 + 7$

$f(-2) = 24 + 13$

$f(-2) = 37$

The remainder is

3

7.

3. Final Answer

37