Short Answer 
To find the equivalent of (dfrac{x^5 – x^3 + 4x^2 – 2}{x^3+3}), perform polynomial long division by dividing the leading terms, multiplying, and subtracting until simplified. The final equivalent expression is (x^2 – 1 + dfrac{x^2 + 1}{x^3 + 3}).
Step 1: Set Up the Long Division
To find the equivalent of the expression (dfrac{x^5 – x^3 + 4x^2 – 2}{x^3+3}), start by arranging the polynomial in the numerator (x^5 – x^3 + 4x^2 – 2) and the denominator (x^3 + 3). Ensure that the terms are in descending order of their power of x. This setup will allow you to perform polynomial long division accurately.
Step 2: Perform the Long Division
Divide the leading term of the numerator by the leading term of the denominator. The calculation will look like this: x^5 ‚à ö‚à ë x^3 = x^2. After finding the term, multiply the entire divisor (x^3 + 3) by x^2 and subtract this from the original numerator. Continue the process until all terms are simplified, revealing the remainder.
Step 3: Write the Final Equivalent Expression
After completing the long division, you will obtain a simpler expression. From our computations, you will find the quotient and remainder. Consequently, the equivalent expression can be assembled as follows: (x^2 – 1 + dfrac{x^2 + 1}{x^3 + 3}). This equation represents the original expression in a simplified form, highlighting the quotient and the remaining fraction.