How do you divide #(5x^3  7x^2 + 14) / (x^2  2)#?
Solution:
Polynomial long division like so
Step 3: Multiply the quotient from step 2, to the divisor and subtract form dividend.
Repeat step 2 and 3 until we can't divide any more.
Q(x) = Quotient R(x) = Remainder d= divisor
#{: (,,5x,+3,,), (,,"","","",""), (x^22,")",5x^3,7x^2,+0x,+14), (,,5x^3,10x^2,,), (,,"","",,), (,,,3x^2,+0x,), (,,,3x^2,6x,), (,,,"","",), (,,,,6x,+14) :}#
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To divide (5x^3  7x^2 + 14) by (x^2  2), you can use long division. Here are the steps:

Divide the first term of the numerator (5x^3) by the first term of the denominator (x^2). This gives you 5x.

Multiply the entire denominator (x^2  2) by the quotient obtained in step 1 (5x). This gives you 5x(x^2  2) = 5x^3  10x.

Subtract the result obtained in step 2 from the numerator (5x^3  7x^2 + 14) to get the new numerator: (7x^2 + 14)  (5x^3  10x) = 7x^2 + 14  5x^3 + 10x.

Repeat steps 13 with the new numerator (7x^2 + 14  5x^3 + 10x) and the denominator (x^2  2).

Divide the first term of the new numerator (7x^2) by the first term of the denominator (x^2). This gives you 7.

Multiply the entire denominator (x^2  2) by the quotient obtained in step 5 (7). This gives you 7(x^2  2) = 7x^2 + 14.

Subtract the result obtained in step 6 from the new numerator (7x^2 + 14  5x^3 + 10x) to get the new numerator: (5x^3 + 10x)  (7x^2 + 14) = 5x^3 + 10x + 7x^2  14.

Repeat steps 17 with the new numerator (5x^3 + 10x + 7x^2  14) and the denominator (x^2  2).

Divide the first term of the new numerator (5x^3) by the first term of the denominator (x^2). This gives you 5x.

Multiply the entire denominator (x^2  2) by the quotient obtained in step 9 (5x). This gives you 5x(x^2  2) = 5x^3 + 10x.

Subtract the result obtained in step 10 from the new numerator (5x^3 + 10x + 7x^2  14) to get the new numerator: (7x^2  14)  (5x^3 + 10x) = 7x^2  14 + 5x^3  10x.

Repeat steps 111 with the new numerator (7x^2  14 + 5x^3  10x) and the denominator (x^2  2).
Continue this process until the degree of the new numerator is less than the degree of the denominator. The final quotient will be the sum of all the quotients obtained in each step.
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