Question

subtract.
\\(\dfrac{3}{2x^2 + 9x + 9} - \dfrac{1}{2x^2 + 7x + 6}\\)
simplify your answer as much as possible.

Explanation:

Step1: Factor denominators

Factor $2x^2+9x+9$: $(2x+3)(x+3)$
Factor $2x^2+7x+6$: $(2x+3)(x+2)$

Step2: Find common denominator

Common denominator: $(2x+3)(x+3)(x+2)$

Step3: Rewrite fractions

$$\frac{3(x+2)}{(2x+3)(x+3)(x+2)} - \frac{1(x+3)}{(2x+3)(x+3)(x+2)}$$

Step4: Combine numerators

$$\frac{3(x+2) - (x+3)}{(2x+3)(x+3)(x+2)}$$

Step5: Simplify numerator

Expand: $3x+6 -x -3 = 2x+3$
$$\frac{2x+3}{(2x+3)(x+3)(x+2)}$$

Step6: Cancel common factors

Cancel $2x+3$ (where $2x+3
eq0$)
$$\frac{1}{(x+3)(x+2)}$$

Step7: Expand denominator (optional)

$$\frac{1}{x^2+5x+6}$$

Answer:

$\frac{1}{(x+3)(x+2)}$ (or $\frac{1}{x^2+5x+6}$)