Answer:
The quotient is [tex]x+3[/tex]
Step-by-step explanation:
We first use the remainder theorem to determine whether [tex]x+7[/tex] is divisible by
[tex]x^2+10x+21[/tex]:
[tex](-7)^2+10(-7)+21\\ = 49 - 70+21 = 0[/tex]
the remainder is zero which means [tex]x^2+10x+21[/tex] is divisible by [tex]x+7[/tex].
Now, we could do polynomial long division to find the quotient, but in this case, it turns out that guessing is easier.
[tex]x^2+10x+21[/tex]
is factored as
[tex]x^2+10x+21 = (x+7)(x+3)[/tex]
which means
[tex]\dfrac{x^2+10x+21}{x+7} =(x+3)[/tex]
which means the quotient is [tex]x+3.[/tex]
