If x+1 and x-+2 are factors of the polynomial $$ax^2 + bx + c,$$ $$\frac{a + b + c}{-a + b + c} = \, ?$$​​

Correct option is D

Given: 

The polynomial $$ax^2 + bx + c$$, will be zero when x = -1 and x = -2 

Step 1: Substitute x = -1 in the polynomials: 

​$$\\a(-1)^2 + b(-1) + c = 0 \\a - b + c = 0$$​​

Step 2: Substitute x = -2 in the polynomial: 

​$$\\a(-2)^2 + b(-2) + c = 0 \\4a - 2b + c = 0 \quad \text{(2)}\\$$​

Step 3: Subtract equation (1) from equation (2):  

​$$\\(4a - 2b + c) - (a - b + c) = 0 \\4a - 2b + c - a + b - c = 0 \\3a - b = 0 \\b = 3a\\$$​​

Step 4: Substitute b = 3a into equation (1): 

a - 3a + c = 0 

-2a + c = 0 

c = 2a 

Step 5: Substitute b = 3a and c = 2a into $$\frac{a + b + c}{-a + b + c}:$$

$$\\a + b + c = a + 3a + 2a = 6a \\-a + b + c = -a + 3a + 2a = 4a \\$$

​$${Step 6: Simplify the ratio: } \\\frac{a + b + c}{-a + b + c} = \frac{6a}{4a} = \frac{3}{2}\\$$​​