Let's determine the sum of the given expression step-by-step:
1. The original expression is:
[tex]\[
\frac{3 x+6}{x^2-x-6}+\frac{2 x}{x^2+x-12}
\][/tex]
2. Factor the denominators:
- [tex]\(x^2 - x - 6 = (x - 3)(x + 2)\)[/tex]
- [tex]\(x^2 + x - 12 = (x - 3)(x + 4)\)[/tex]
Replace the denominators in the original fractions:
[tex]\[
\frac{3 x+6}{(x - 3)(x + 2)}+\frac{2 x}{(x - 3)(x + 4)}
\][/tex]
3. Rewrite the first term's numerator so the denominators can be the same:
[tex]\[
\frac{3(x+2)}{(x - 3)(x + 2)}+\frac{2 x}{(x - 3)(x + 4)}
\][/tex]
4. Make the denominators common for addition:
[tex]\[
\frac{3(x+2)}{(x - 3)(x + 4)} + \frac{2 x}{(x - 3)(x + 4)}
\][/tex]
5. Combine the numerators over the common denominator:
[tex]\[
\frac{3(x+2) + 2x}{(x - 3)(x + 4)}
\][/tex]
6. Simplify the numerator:
[tex]\[
3(x+2) + 2x = 3x + 6 + 2x = 5x + 6
\][/tex]
7. Write the final simplified expression:
[tex]\[
\frac{5x + 6}{(x - 3)(x + 4)}
\][/tex]
So, the steps in order are:
1. [tex]\(\frac{3(x+2)}{(x+2)(x-3)}+\frac{2 x}{(x-3)(x+4)}\)[/tex]
2. [tex]\(\frac{3(x+2)}{(x - 3)(x + 4)} + \frac{2 x}{(x - 3)(x + 4)}\)[/tex]
3. [tex]\(\frac{(3 x+6)+2 x}{(x-3)(x+4)}\)[/tex]
4. [tex]\(\frac{5 x+12}{(x-3)(x+4)}\)[/tex]
