Simplify: \( \frac{x}{x-3}-\frac{3}{x+4}+\frac{7}{x^{2}+x-12} \) Rewrite the expression using the LCD:

To simplify the expression \( \frac{x}{x-3} - \frac{3}{x+4} + \frac{7}{x^2+x-12} \), we first need to factor the denominator of the third term. The expression \( x^2 + x - 12 \) factors into \( (x-3)(x+4) \). Now, the least common denominator (LCD) for the fractions will be \( (x-3)(x+4) \). Next, we will rewrite each term with the LCD: 1. For \( \frac{x}{x-3} \), we multiply the numerator and denominator by \( (x+4) \): \[ \frac{x(x+4)}{(x-3)(x+4)} = \frac{x^2 + 4x}{(x-3)(x+4)} \] 2. For \( -\frac{3}{x+4} \), we multiply the numerator and denominator by \( (x-3) \): \[ -\frac{3(x-3)}{(x+4)(x-3)} = \frac{-3x + 9}{(x-3)(x+4)} \] 3. For \( \frac{7}{x^2+x-12} \), we can rewrite it directly as: \[ \frac{7}{(x-3)(x+4)} \] Putting it all together, we have: \[ \frac{x^2 + 4x - 3x + 9 + 7}{(x-3)(x+4)} \] This simplifies the numerator to: \[ \frac{x^2 + x + 16}{(x-3)(x+4)} \] So, the simplified expression is: \[ \frac{x^2 + x + 16}{(x-3)(x+4)} \]