x^2+10x+21/x+3? If you meant (x^2+10x+21)/(x+3), please note that those parentheses are essential.
If you wish to divide x^2+10x+21 by (x+3) using synth. div., use the divisor -3:
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-3 / 1 10 21
-3 -21
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1 7 0
Since the remainder is 0, (x+3) is a divisor of x^2+10x+21, and the other factor is (1x + 7).