Polynomial long division works the same way as regular long division of decimal numbers. THE SAME WAY. Abbreviations are not done so you can see the entire part of each step.
Your dividend is , because you must account for all degrees of x up to the degree of the dividend. Your divisor is and you might want but not need to use it as .
You will expect that divisor is less than dividend, and ask yourself what is the quotient of the leading terms. This will be the first partial quotient to start the process. I will not explain the rest but will show just the start...
; indicate this partial quotient. Multiply this by the divisor, and put result in proper positions.
2x^2
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2x^2+1 | 4x^4 6x^3 0x^2 3x -1
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| 4x^4 0x^2 2x^2
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Do the necessary subtraction.
2x^2
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2x^2+1 | 4x^4 6x^3 0x^2 3x -1
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| 4x^4 0x^2 2x^2
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0 6x^2 -2x^2
Bring down the next term, or terms, and continue. You would then be asking yourself, what is .