Certainly! Let's go through the steps required to determine the sum of the two expressions in the correct order and place the tiles accordingly:
### Step-by-Step Solution:
1. Factorize the expressions in the denominators and numerators where possible:
Given expression:
[tex]\[ \frac{3x + 6}{x^2 - x - 6} + \frac{2x}{x^2 + x - 12} \][/tex]
Factorize the denominators and the numerator of the first term:
[tex]\[ x^2 - x - 6 = (x - 3)(x + 2) \][/tex]
[tex]\[ x^2 + x - 12 = (x - 3)(x + 4) \][/tex]
[tex]\[ 3x + 6 = 3(x + 2) \][/tex]
Transformed expression:
[tex]\[ \frac{3(x + 2)}{(x - 3)(x + 2)} + \frac{2x}{(x - 3)(x + 4)} \][/tex]
2. Simplify the first term:
Cancel out the common factors in the numerator and denominator of the first term:
[tex]\[ \frac{3(x + 2)}{(x - 3)(x + 2)} = \frac{3}{(x - 3)} \][/tex]
Simplified expression:
[tex]\[ \frac{3}{(x - 3)} + \frac{2x}{(x - 3)(x + 4)} \][/tex]
3. Find a common denominator for combining the fractions:
Combine the fractions using the common denominator [tex]\((x - 3)(x + 4)\)[/tex]:
[tex]\[ \frac{3}{(x - 3)} = \frac{3(x + 4)}{(x - 3)(x + 4)} \][/tex]
Expression with a common denominator:
[tex]\[ \frac{3(x + 4)}{(x - 3)(x + 4)} + \frac{2x}{(x - 3)(x + 4)} \][/tex]
4. Combine the numerators:
Combine the numerators over the common denominator:
[tex]\[ \frac{3(x + 4) + 2x}{(x - 3)(x + 4)} = \frac{3x + 12 + 2x}{(x - 3)(x + 4)} \][/tex]
Simplify the numerator:
[tex]\[ \frac{5x + 12}{(x - 3)(x + 4)} \][/tex]
So, the collection of steps in order would be:
1. [tex]\(\square\)[/tex]
2. [tex]\(\frac{5x + 12}{(x - 3)(x + 4)}\)[/tex]
3. [tex]$\downarrow$[/tex]
4. [tex]\(\frac{3(x+2)}{(x+2)(x-3)}+\frac{2x}{(x-3)(x+4)}\)[/tex]
5. [tex]$\downarrow$[/tex]
6. [tex]\(\frac{3}{(x-3)}+\frac{2x}{(x-3)(x+4)}\)[/tex]
7. [tex]$\downarrow$[/tex]
8. [tex]\(\frac{3(x+4)}{(x-3)(x+4)}+\frac{2x}{(x-3)(x+4)}\)[/tex]
Note: The tile "[tex]\(\frac{5 x+6}{x^2-x+12}\)[/tex]" is not used because it does not align with the correct factorization and simplification steps we need.
