Question

8 multiple choice 1 point
given $f(x)=x^2 + 8x + 15$ and $g(x)=x + 2$; then $\frac{f(x)}{g(x)}=x + 6+\frac{c}{x+2}$
what is the value of c?
2
8
3
6
clear my selection

Explanation:

Step1: Set up equality of expressions

$$\frac{f(x)}{g(x)} = x + 6 + \frac{c}{x+2}$$
Substitute $f(x)=x^2+8x+15$ and $g(x)=x+2$:
$$\frac{x^2+8x+15}{x+2} = x + 6 + \frac{c}{x+2}$$

Step2: Eliminate denominator

Multiply all terms by $x+2$:
$$x^2+8x+15 = (x+6)(x+2) + c$$

Step3: Expand right-hand side

Calculate $(x+6)(x+2)$:
$$(x+6)(x+2) = x^2+2x+6x+12 = x^2+8x+12$$
So the equation becomes:
$$x^2+8x+15 = x^2+8x+12 + c$$

Step4: Solve for c

Subtract $x^2+8x+12$ from both sides:
$$15 - 12 = c$$
$$c=3$$

Answer:

C. 3