Add the rational expressions, and then simplify. \[ \frac{6}{a+1}+\frac{7}{a-3} \]

Step 1 ：Given the rational expressions \(\frac{6}{a+1}\) and \(\frac{7}{a-3}\).

Step 2 ：To add these two rational expressions, we need to find a common denominator. The common denominator of the two fractions is the product of the two denominators, i.e., \((a+1)(a-3)\).

Step 3 ：We rewrite each fraction with the common denominator and add the numerators.

Step 4 ：The simplified form of the given rational expressions is \(\boxed{\frac{13a - 11}{(a - 3)(a + 1)}}\)