$$ f(x)=x^{2}+5 x+6 $$ $$ g(x)=x+2 $$ Find $$ \left(\frac{f}{g} ight)(x) $$ $$ \left(\frac{f}{g} ight)(x)=\square $$ The domain of $$ \left(\frac{f}{g} ight)(x) $$ is $$ x
eq \square $$

Question

$$ f(x)=x^{2}+5 x+6 $$ $$ g(x)=x+2 $$ Find $$ \left(\frac{f}{g}ight)(x) $$ $$ \left(\frac{f}{g}ight)(x)=\square $$ The domain of $$ \left(\frac{f}{g}ight)(x) $$ is $$ x 
eq \square $$

UpStudy AI · Accepted Answer

1. We are given 
   $$
   f(x)=x^{2}+5x+6 \quad \text{and} \quad g(x)=x+2.
   $$

2. To find $$\left(\frac{f}{g}\right)(x)$$, we compute 
   $$
   \frac{f(x)}{g(x)}=\frac{x^{2}+5x+6}{x+2}.
   $$

3. Factor the numerator:
   $$
   x^{2}+5x+6=(x+2)(x+3).
   $$

4. Substitute the factored form into the expression:
   $$
   \frac{f(x)}{g(x)}=\frac{(x+2)(x+3)}{x+2}.
   $$

5. Cancel the $$(x+2)$$ terms, noting that the cancellation is valid only if $$x+2\neq0$$ (i.e., $$x\neq-2$$):
   $$
   \frac{(x+2)(x+3)}{x+2}=x+3 \quad \text{for} \quad x\neq-2.
   $$

6. Therefore, 
   $$
   \left(\frac{f}{g}\right)(x)=x+3,
   $$
   and the domain of $$\left(\frac{f}{g}\right)(x)$$ is 
   $$
   x\neq-2.
   $$
$$ \left(\frac{f}{g}\right)(x) = x + 3 $$ The domain of $$\left(\frac{f}{g}\right)(x)$$ is $$ x \neq -2 $$.

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