We are given the equation $x^4 - 2x^3 - 3x^2 + 4x + 4 = 0$. To find its roots in a simple way, we can apply synthetic division and factorization techniques. Let's try to identify potential rational roots using the Rational Root Theorem, and then simplify step by step.

Step 1: Use the Rational Root Theorem

The Rational Root Theorem suggests possible rational roots as factors of the constant term (4) divided by factors of the leading coefficient (1). Therefore, the possible rational roots are:

$\pm 1, \pm 2, \pm 4$

Step 2: Test the possible roots

We will test these roots using synthetic division or direct substitution to find one that works.

• Testing $x = 1$:

$1^4 - 2(1^3) - 3(1^2) + 4(1) + 4 = 1 - 2 - 3 + 4 + 4 = 4$

So, $x = 1$ is not a root.

• Testing $x = -1$:

$(-1)^4 - 2(-1)^3 - 3(-1)^2 + 4(-1) + 4 = 1 + 2 - 3 - 4 + 4 = 0$

Thus, $x = -1$ is a root.

Step 3: Perform synthetic division

Since $x = -1$ is a root, we can divide the polynomial by $x + 1$ to simplify it.

Let's perform synthetic division of $x^4 - 2x^3 - 3x^2 + 4x + 4$ by $x + 1$:

-1 & 1 & -2 & -3 & 4 & 4 \\ & & -1 & 3 & 0 & -4 \\ \hline & 1 & -3 & 0 & 4 & 0 \\ \end{array}$$ The quotient is $$x^3 - 3x^2 + 4$$, and the remainder is