We are dividing [tex]3x^2-2x+7[/tex] by (x+1).
According to the "Division algorithm":
P(x)=D(x)*Q(x)+R(x),
where P(x) is the dividend polynomial, D(x) is the divisor polynomial, Q(x) is the quotient polynomial and R(x) the remainder which must be of smaller degree than the divisor.
so
[tex]3x^2-2x+7=(x + 1)Q(x)+c[/tex]
R(x)=c, a constant, because the divisor is a linear.
also notice that Q(x) must be a linear of the form 3x+a:
[tex]3x^2-2x+7=(x + 1)(3x+a)+c\\\\3x^2-2x+7=3x^2+ax+3x+a+c\\\\3x^2-2x+7=3x^2+(a+3)x+(a+c)[/tex]
comparing the coefficients, we have:
-2=a+3 and 7=a+c
a=-5, then 7=-5+c, so c=-12
Answer: the quotient is 3x-5
