Question

what should be subtracted from 4 x cube + 8 x square + 8 x + 7 to make it divisible by 2 x squared minus 5 + 1

Answer 1

2x²-5+1=0 (suppose)
2x²-4=0
2x²=4
x²=2
x=√2 (eq¹)

From Remainder Theorem,
4x³+8x²+8x+7 =0 (suppose)
=8√2+16+8√2+7=0
=16√2+21. ≠0
We must subtract 16√2+21 from the polynomial to make it divisible by the divident.

Answer 2

$16 \sqrt[2] + 21$

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