Question: User: top-left: Add and simplify:
\[\frac{x^2+4x}{x^2+3x-54}-\frac{4}{x-6}\]
bottom-left: Hint: Find the LCD by factoring the first denominator, multiply the first fraction by
\[\frac{x+9}{x+9}\]
, distribute the -4 to \((x+9)\), combine like terms across the top, and then simplify
the remaining fraction
1. **State the problem:** Simplify the expression $$\frac{x^2+4x}{x^2+3x-54}-\frac{4}{x-6}$$.
2. **Factor the denominator of the first fraction:**
$$x^2+3x-54 = (x+9)(x-6)$$
3. **Rewrite the expression with factored denominator:**
$$\frac{x^2+4x}{(x+9)(x-6)} - \frac{4}{x-6}$$
4. **Find the least common denominator (LCD):**
The LCD is $$(x+9)(x-6)$$.
5. **Rewrite the second fraction with the LCD:**
Multiply numerator and denominator by $$(x+9)$$:
$$\frac{4}{x-6} = \frac{4(x+9)}{(x-6)(x+9)}$$
6. **Rewrite the expression with common denominator:**
$$\frac{x^2+4x}{(x+9)(x-6)} - \frac{4(x+9)}{(x+9)(x-6)} = \frac{x^2+4x - 4(x+9)}{(x+9)(x-6)}$$
7. **Distribute and combine like terms in the numerator:**
$$x^2 + 4x - 4x - 36 = x^2 - 36$$
8. **Recognize the numerator as a difference of squares:**
$$x^2 - 36 = (x-6)(x+6)$$
9. **Simplify the fraction:**
$$\frac{(x-6)(x+6)}{(x+9)(x-6)} = \frac{\cancel{(x-6)}(x+6)}{(x+9)\cancel{(x-6)}} = \frac{x+6}{x+9}$$
**Final answer:**
$$\frac{x+6}{x+9}$$