## Simplifying Polynomials: (x² + 1 - 4x³ + 3x⁴) + (-x² - 2x⁴ + x³ + 4)

This article explores the process of simplifying the given polynomial expression: **(x² + 1 - 4x³ + 3x⁴) + (-x² - 2x⁴ + x³ + 4)**. We will break down the steps involved in combining like terms and arriving at the simplified form.

### Understanding Polynomials

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. The **degree** of a polynomial is determined by the highest exponent of the variable. In our expression, the highest exponent is 4, making it a fourth-degree polynomial.

### Combining Like Terms

To simplify the expression, we need to combine **like terms**. Like terms are terms with the same variable and exponent. Let's group the terms by their variable and exponent:

**x⁴ terms:**3x⁴ - 2x⁴**x³ terms:**-4x³ + x³**x² terms:**x² - x²**Constant terms:**1 + 4

### Performing the Operations

Now, we can combine the coefficients of each group:

**x⁴ terms:**(3 - 2)x⁴ = x⁴**x³ terms:**(-4 + 1)x³ = -3x³**x² terms:**(1 - 1)x² = 0**Constant terms:**1 + 4 = 5

### The Simplified Expression

Finally, we combine the results to obtain the simplified polynomial:

**x⁴ - 3x³ + 5**

Therefore, the simplified form of the expression **(x² + 1 - 4x³ + 3x⁴) + (-x² - 2x⁴ + x³ + 4)** is **x⁴ - 3x³ + 5**.