Question 1037649
<pre><b>
-1|-2 -5 4 2 3 
  |___________ 
   -2 -3 7

That's not complete.  It should be

-1|-2 -5  4  2 3 
  |<u>    2  3 -7 5</u> 
   -2 -3  7 -5 8

Synthetic division is a shortcut for long division
of a polynomial by a binomial of the form x±A:

    <u>    -2x³-3x²+7x-5</u>+{{{8/(x+1)}}}
x+1)-2x<sup>4</sup>-5x³+4x²+2x+3
    <u>-2x4-2x³</u>
        -3x³+4x²
        <u>-3x³-3x²</u>
             7x²+2x
             <u>7x²+7x</u>
                -5x+3
                <u>-5x-5</u>
                    8       

The remainder when you divide a polynomial by x+1 is always 
the same as what you get if you substitute (-1) in the polynomial.

-2(-1)<sup>4</sup>-5(-1)³+4(-1)²+2(-1)+3
-2(1)-5(-1)+4(1)-2+3
-2+5+4-2+3
   8

Notice that the 8 remainder when you divide by x+1 
using long division or the opposite sign -1 with
synthetic division is the same answer as if you
had substituted -1 into the polynomial.  So f(-1)=8

Edwin</pre></b>