Ireally need consider the following division of polynomials. (x^4+x^3+7x^2-6x+8)/ (x^2+2x+8) a) use long division to determine the quotient of the polynomials. show all of your work for full credit. b) use mathematical methods to prove your answer. show all of your work for full credit.

x²-x+1

Step-by-step explanation:

We write the dividend, x⁴+x³+7x²-6x+8, under the division box.  We write the divisor, x²+2x+8, outside the box to the left.

We see how many times x² goes into x⁴.  It will go x² times.  We write this on the line above the dividend.  We multiply back through;

x²(x²+2x+8) = x⁴+2x³+8x²

We write this under the first three terms of the dividend.  We then subtract:

(x⁴+x³+7x²)-(x⁴+2x³+8x²) = -1x³-1x²

This goes below the subtraction.  We bring down the next term, -6x.

Next we see how many times x² goes into -1x³.  It goes -x times; we write this beside the x² in the quotient.  We multiply back through:

-x(x²+2x+8) = -x³-2x²-8x

This goes below the -1x³-1x²-6x.  We subtract the two:

(-1x³-1x²-6x)-(-x³-2x²-8x) = -1x³+1x³-1x²+2x²-6x+8x = 1x²+2x

We write this below the last line in the work.  We bring down the last term, 8.

Now we see how many times x² goes into 1x².  It goes 1 time; since this is positive, we write +1 beside the answer we have at the top.  Multiply back through:

1(x²+2x+8) = x²+2x+8

Now we subtract:

(x²+2x+8)-(x²+2x+8) = 0

This makes our answer x²-x+1