Question

Verify that $\left(x^2-1\right)+\frac{x^2}{x^4+3x^2+4}$is a valid result for the polynomial long-division calculation performed to divide the polynomial $x^6+2x^4+2x^2-4$by the polynomial$x^4+3x^2+4$

Short answer

The given result$\left(x^2-1\right)+\frac{x^2}{x^4+3x^2+4}$is verified.

Step-by-step solution

Step 1. Given Information

The given improper function is$\frac{x^6+2x^4+2x^2-4}{x^4+3x^2+4}$

Step 2. Calculation

The long division methof is shown below,

So, the improper function is written as a sum of an integer and a proper function below,

$\frac{x^6+2x^4+2x^2-4}{x^4+3x^2+4}=\left(x^2-1\right)+\frac{x^2}{x^4+3x^2+4}$

Thus, the result is verified.