1) \( \frac{3}{a-4}+\frac{2}{a+3} \cdots \frac{21}{a^{2}-a-12} \)

We wish to simplify \[ \frac{3}{a-4}+\frac{2}{a+3}+\frac{21}{a^{2}-a-12}. \] Notice that the quadratic in the denominator of the third fraction factors: \[ a^{2}-a-12=(a-4)(a+3). \] Thus the expression becomes \[ \frac{3}{a-4}+\frac{2}{a+3}+\frac{21}{(a-4)(a+3)}. \] To combine these three fractions into a single fraction, we use the common denominator \((a-4)(a+3)\). 1. Write the first term over the common denominator: \[ \frac{3}{a-4}=\frac{3(a+3)}{(a-4)(a+3)}. \] 2. Write the second term over the common denominator: \[ \frac{2}{a+3}=\frac{2(a-4)}{(a-4)(a+3)}. \] 3. The third term already has the denominator \((a-4)(a+3)\). Now, the sum is \[ \frac{3(a+3)+2(a-4)+21}{(a-4)(a+3)}. \] Let’s simplify the numerator by expanding and combining like terms: - Expand: \(3(a+3)=3a+9\) and \(2(a-4)=2a-8\). - Add: \[ 3a+9+2a-8+21=(3a+2a)+ (9-8+21)=5a+22. \] Thus, the simplified expression is \[ \frac{5a+22}{(a-4)(a+3)}. \]