Simplify the following algebraic expression \( \frac{x+3}{x^{2}-7 x+12}-\frac{x-3}{x^{2}-9} \)

To simplify the given algebraic expression \( \frac{x+3}{x^{2}-7 x+12}-\frac{x-3}{x^{2}-9} \), we first factor the quadratic expressions in the denominators. 1. The first denominator: \( x^{2} - 7x + 12 = (x - 3)(x - 4) \) 2. The second denominator: \( x^{2} - 9 = (x - 3)(x + 3) \) Now, we can rewrite the expression: \[ \frac{x + 3}{(x - 3)(x - 4)} - \frac{x - 3}{(x - 3)(x + 3)} \] Since both fractions have a common factor of \( x - 3 \) in the second denominator, we will combine these fractions over a common denominator, which is \( (x - 3)(x - 4)(x + 3) \). The first fraction needs to be multiplied by \((x + 3)\) to get a common denominator: \[ \frac{(x + 3)(x + 3)}{(x - 3)(x - 4)(x + 3)} \] Rearranging the second fraction: \[ \frac{-(x - 3)(x - 4)}{(x - 3)(x + 3)(x - 4)} \] Now we can combine the two fractions: \[ \frac{(x + 3)^2 - (x - 3)(x - 4)}{(x - 3)(x - 4)(x + 3)} \] Expanding and simplifying the numerator: 1. Expand \((x + 3)^2\) to get \(x^2 + 6x + 9\). 2. Expand \((x - 3)(x - 4)\) to get \(x^2 - 7x + 12\). Combining these gives: \[ x^2 + 6x + 9 - (x^2 - 7x + 12) = x^2 + 6x + 9 - x^2 + 7x - 12 = 13x - 3 \] So, our expression becomes: \[ \frac{13x - 3}{(x - 3)(x - 4)(x + 3)} \] This is your simplified expression.