Let `alpha,beta` be the roots of the quadratic equation `ax^2+bx+c=0` then the roots of the equation `a(x+1)^2+b(x+1)(x-2)+c(x-2)^2=0` are:-

Options

A. `(2 alpha +1)/(alpha -1) , (2 beta+1)/(beta -1)`
B. `(2 alpha -1)/(alpha +1) , (2 beta)/(alpha +1), (2 beta -1)/(beta +1)`
C. `(alpha +1)/(alpha -2) , (beta +1)/(beta -2)`
D. `(2 alpha +3)/(alpha -1), (2 beta+3)/(beta -1)`

Answer

To solve the problem, we need to find the roots of the equation \( a(x+1)^2 + b(x+1)(x-2) + c(x-2)^2 = 0 \) given that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \).

### Step 1: Expand the equation
We start by expanding the given equation:

\[ a(x+1)^2 + b(x+1)(x-2) + c(x-2)^2 = 0 \]

Expanding each term:

1. \( a(x+1)^2 = a(x^2 + 2x + 1) = ax^2 + 2ax + a \)
2. \( b(x+1)(x-2) = b(x^2 - 2x + x - 2) = b(x^2 - x - 2) = bx^2 - bx - 2b \)
3. \( c(x-2)^2 = c(x^2 - 4x + 4) = cx^2 - 4cx + 4c \)

Now, combine all these expansions:

\[ (ax^2 + 2ax + a) + (bx^2 - bx - 2b) + (cx^2 - 4cx + 4c) = 0 \]

### Step 2: Combine like terms
Now, we combine the coefficients of \( x^2 \), \( x \), and the constant term:

- Coefficient of \( x^2 \): \( a + b + c \)
- Coefficient of \( x \): \( 2a - b - 4c \)
- Constant term: \( a - 2b + 4c \)

Thus, the equation becomes:

\[ (a + b + c)x^2 + (2a - b - 4c)x + (a - 2b + 4c) = 0 \]

### Step 3: Identify the new quadratic equation
We can denote the new quadratic equation as:

\[ A x^2 + B x + C = 0 \]

where:
- \( A = a + b + c \)
- \( B = 2a - b - 4c \)
- \( C = a - 2b + 4c \)

### Step 4: Find the roots using the quadratic formula
The roots of the quadratic equation \( Ax^2 + Bx + C = 0 \) can be found using the quadratic formula:

\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

### Step 5: Substitute \( A \), \( B \), and \( C \)
Substituting the values of \( A \), \( B \), and \( C \):

1. Calculate \( B^2 - 4AC \):
\[ B^2 = (2a - b - 4c)^2 \]
\[ 4AC = 4(a + b + c)(a - 2b + 4c) \]

2. Substitute these into the quadratic formula to find the roots.

### Step 6: Conclusion
The roots of the equation \( a(x+1)^2 + b(x+1)(x-2) + c(x-2)^2 = 0 \) can be expressed in terms of \( \alpha \) and \( \beta \) based on the relationships derived from the coefficients.