y2+25=y2+0y+25 is represented by the sequence 1, 0, 25.
y+5 is represented by the sequence 1, 5.
Write out like long division of integers and proceed similarly:
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Write 1, 0, 25 under the bar as the dividend and 1, 5 to the left as the divisor.
Identify 1 as the first term of the quotient - writing it above the bar - choosing it to cause the leading terms to match when the divisor is multiplied by it.
Write out 1×(1,5) under the dividend and subtract it to get the first term −5 of a remainder.
Bring down the next term 25 of the dividend alongside it.
Identify −5 as the second term of the quotient - writing above the bar - choosing it to cause the leading terms to match when the divisor is multiplied by it.
Write out −5×(1,5) under the remainder and subtract it to get the remainder 50.
We stop with this remainder since there are not enough terms remaining to make a sequence with enough terms to be divisible by the divisor.
So we have found:
y2+25y+5=(y−5)+50y+5
or if you prefer:
y2+25=(y+5)(y−5)+50
