If `alpha` and `beta` are the roots of `ax^2 + bx + c = 0`, find the equation whose roots are `alpha +2` and `beta +2`.

Answer

To find the equation whose roots are \( \alpha + 2 \) and \( \beta + 2 \), we can follow these steps:

### Step 1: Identify the roots of the original equation
The original quadratic equation is given by:
\[ ax^2 + bx + c = 0 \]
The roots of this equation are \( \alpha \) and \( \beta \).

### Step 2: Use the relationships for the sum and product of the roots
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 3: Find the new roots
The new roots we are interested in are:
\[ \alpha + 2 \quad \text{and} \quad \beta + 2 \]

### Step 4: Calculate the sum of the new roots
The sum of the new roots is:
\[ (\alpha + 2) + (\beta + 2) = \alpha + \beta + 4 \]
Substituting the value of \( \alpha + \beta \):
\[ \alpha + \beta + 4 = -\frac{b}{a} + 4 \]

### Step 5: Calculate the product of the new roots
The product of the new roots is:
\[ (\alpha + 2)(\beta + 2) = \alpha \beta + 2\alpha + 2\beta + 4 \]
Substituting the values of \( \alpha + \beta \) and \( \alpha \beta \):
\[ \alpha \beta + 2(\alpha + \beta) + 4 = \frac{c}{a} + 2\left(-\frac{b}{a}\right) + 4 \]
This simplifies to:
\[ \frac{c}{a} - \frac{2b}{a} + 4 = \frac{c - 2b + 4a}{a} \]

### Step 6: Form the new quadratic equation
Using the sum and product of the new roots, we can form the new quadratic equation:
\[ x^2 - \text{(sum of roots)} \cdot x + \text{(product of roots)} = 0 \]
Substituting the values we found:
\[ x^2 - \left(-\frac{b}{a} + 4\right)x + \frac{c - 2b + 4a}{a} = 0 \]

### Step 7: Multiply through by \( a \) to eliminate the fraction
Multiplying the entire equation by \( a \):
\[ ax^2 + \left(b - 4a\right)x + (c - 2b + 4a) = 0 \]

### Final Answer
The equation whose roots are \( \alpha + 2 \) and \( \beta + 2 \) is:
\[ ax^2 + (b - 4a)x + (c - 2b + 4a) = 0 \]