can you solve:

(x+1)*x^3 + x*(x^3 +1) = (mod x^4 +x+1)
x*x^3 + (x+1)*(x^3 +1) = (mod x^4 +x+1)
(x+1) + x*(x^3 + x^2) = (mod x^4 +x+1)
x + (x+1)*(x^3 + x^2) = (mod x^4 +x+1)

The expressions you provided seem to involve modular arithmetic with respect to \(x^4 + x + 1\). To clarify, you want to simplify or solve these expressions under the modulus \(x^4 + x + 1\). Let's proceed by simplifying each of these expressions step by step, modulo \(x^4 + x + 1\).

### Expression 1:
\[
(x+1) \cdot x^3 + x \cdot (x^3 + 1) = \text{mod } (x^4 + x + 1)
\]
First, expand both terms on the left-hand side:
\[
(x+1) \cdot x^3 = x^4 + x^3
\]
\[
x \cdot (x^3 + 1) = x^4 + x
\]
Now, sum the expanded terms:
\[
x^4 + x^3 + x^4 + x = 2x^4 + x^3 + x
\]
Next, apply the modulus \(x^4 + x + 1\):
- Notice that \(x^4 \equiv -x - 1 \mod (x^4 + x + 1)\).
Thus, substitute \(x^4\) with \(-x - 1\):
\[
2(-x - 1) + x^3 + x = -2x - 2 + x^3 + x
\]
Simplify:
\[
x^3 - x - 2
\]
So, the first expression simplifies to:
\[
x^3 - x - 2 \mod (x^4 + x + 1)
\]

### Expression 2:
\[
x \cdot x^3 + (x+1) \cdot (x^3 + 1) = \text{mod } (x^4 + x + 1)
\]
Expand both terms:
\[
x \cdot x^3 = x^4
\]
\[
(x+1) \cdot (x^3 + 1) = x^4 + x^3 + x^3 + 1 = x^4 + 2x^3 + 1
\]
Sum the terms:
\[
x^4 + x^4 + 2x^3 + 1 = 2x^4 + 2x^3 + 1
\]
Apply the modulus \(x^4 + x + 1\):
- Substitute \(x^4 \equiv -x - 1\):
\[
2(-x - 1) + 2x^3 + 1 = -2x - 2 + 2x^3 + 1
\]
Simplify:
\[
2x^3 - 2x - 1
\]
Thus, the second expression simplifies to:
\[
2x^3 - 2x - 1 \mod (x^4 + x + 1)
\]

### Expression 3:
\[
(x+1) + x \cdot (x^3 + x^2) = \text{mod } (x^4 + x + 1)
\]
Expand:
\[
x \cdot (x^3 + x^2) = x^4 + x^3
\]
Thus, the equation becomes:
\[
(x + 1) + x^4 + x^3
\]
Apply the modulus \(x^4 + x + 1\):
\[
(x + 1) + (-x - 1) + x^3 = x^3
\]
Thus, the third expression simplifies to:
\[
x^3 \mod (x^4 + x + 1)
\]

### Expression 4:
\[
x + (x+1) \cdot (x^3 + x^2) = \text{mod } (x^4 + x + 1)
\]
Expand:
\[
(x+1) \cdot (x^3 + x^2) = x^4 + x^3 + x^3 + x^2 = x^4 + 2x^3 + x^2
\]
Thus, the equation becomes:
\[
x + x^4 + 2x^3 + x^2
\]
Apply the modulus \(x^4 + x + 1\):
\[
x + (-x - 1) + 2x^3 + x^2 = 2x^3 + x^2 - 1
\]
Thus, the fourth expression simplifies to:
\[
2x^3 + x^2 - 1 \mod (x^4 + x + 1)
\]

### Final Simplifications:
1. \(x^3 - x - 2\)
2. \(2x^3 - 2x - 1\)
3. \(x^3\)
4. \(2x^3 + x^2 - 1\)

Would you like further details on how these were derived or how to proceed? Here are some related questions:

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1. How does modular arithmetic apply to polynomials?
2. What are the practical applications of polynomial modular arithmetic?
3. Can you explain the process of polynomial division in more detail?
4. How does polynomial factoring work in modular arithmetic?
5. What are some common problems where polynomial modular reduction is used?

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**Tip**: In polynomial modular arithmetic, terms of degree higher than the modulus are replaced using the modulus equation, reducing them to lower degree equivalents.

(x+1)*x^3 + x*(x^3 +1) = (mod x^4 +x+1)
x*x^3 + (x+1)*(x^3 +1) = (mod x^4 +x+1)
(x+1) + x*(x^3 + x^2) = (mod x^4 +x+1)
x + (x+1)*(x^3 + x^2) = (mod x^4 +x+1)

Learn how to solve and simplify polynomial expressions under modulus x^4 + x + 1 using modular arithmetic. This guide covers step-by-step expansions, modulus reductions, and practical polynomial modular techniques. Suitable for undergraduate students studying algebra or computer science.

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