### Theory:

Let us discuss the addition of two rational expressions in two cases.

Case 1: Like Denominators

If the rational numbers are of the form $$\frac{p(x)}{q(x)}$$ and $$\frac{r(x)}{q(x)}$$, the addition is performed as follows.

Working Rule:
Step 1: Add the numerators $$p(x)$$ and $$r(x)$$.

Step 2: Write the sum of the numerators found in the previous step over the common denominator $$q(x)$$.

Step 3: Reduce the resulting rational expression $$\frac{p(x) + r(x)}{q(x)}$$ into its lowest form.
Example:
Add: $$\frac{2x + 3}{x + 1}$$ and $$\frac{5x + 4}{x + 1}$$

Solution:

$$2x + 3$$ $$+$$ $$5x + 4$$ $$=$$ $$(2 + 5)x + (3 + 4)$$

$$=$$ $$7x + 7$$

Step 2: Write the sum of the numerators found in the previous step over the common denominator.

$$\frac{2x + 3}{x + 1}$$ $$+$$ $$\frac{5x + 4}{x + 1}$$ $$=$$ $$\frac{7x + 7}{x + 1}$$

Step 3: Reduce the resulting rational expression into its lowest form.

$$\frac{7x + 7}{x + 1}$$ $$=$$ $$\frac{7(x + 1)}{x + 1}$$

$$=$$ $\frac{7\overline{)\left(x+1\right)}}{\overline{)\left(x+1\right)}}$

$$=$$ $$7$$

Therefore, $$\frac{2x + 3}{x + 1}$$ $$+$$ $$\frac{5x + 4}{x + 1}$$ $$=$$ $$7$$.
Case 2: Unlike Denominators

If the rational numbers are of the form $$\frac{p(x)}{q(x)}$$ and $$\frac{r(x)}{s(x)}$$, the addition is performed as follows.

Working Rule:
Step 1: Determine the Least Common Multiple of the denominator $$q(x)$$ and $$s(x)$$.

Step 2: Rewrite each fraction as an equivalent fraction with the $$LCM$$ obtained in the previous step by multiplying both the numerators and denominator of each expression by any factors needed to obtain the $$LCM$$.

Step 3: Follow the same steps given for doing addition of the rational expression with like denominators.
Example:
Add: $$\frac{-6}{x + 1}$$ and $$\frac{3x}{x}$$

Solution:

Step 1: Determine the Least Common Multiple of the denominator.

$$LCM \left(x+1, x \right)$$ $$=$$ $$x(x+1)$$

Step 2: Rewrite each fraction as an equivalent fraction with the $$LCM$$ obtained in the previous step by multiplying both the numerators and denominator of each expression by any factors needed to obtain the $$LCM$$.

$$\frac{-6}{x + 1}$$ $$+$$ $$\frac{3x}{x}$$ $$=$$ $$\frac{-6 \times x}{(x + 1) \times x}$$ $$+$$ $$\frac{3x \times (x+1)}{x \times (x+1)}$$

$$=$$ $$\frac{-6x}{x(x + 1)}$$ $$+$$ $$\frac{3x^2 + 3x}{x(x+1)}$$

Step 3: Follow the same steps given for doing addition of the rational expression with like denominators.

$$\frac{-6x}{x(x + 1)}$$ $$+$$ $$\frac{3x^2 + 3x}{x(x+1)}$$ $$=$$ $$\frac{3x^2 + (-6 + 3)x}{x(x + 1)}$$

$$=$$ $$\frac{3x^2 - 3x}{x(x + 1)}$$

$$=$$ $$\frac{3x(x - 1)}{x(x + 1)}$$

$$=$$ $\frac{3\overline{)x}\left(x-1\right)}{\overline{)x}\left(x+1\right)}$

$$=$$ $$\frac{3(x-1)}{x+1}$$

Therefore, $$\frac{-6}{x + 1}$$ $$+$$ $$\frac{3x}{x }$$ $$=$$ $$\frac{3(x-1)}{x+1}$$.