If `alpha and beta` are the roots of the equation `2x^(2)+3x+2=0`, find the equation whose roots are `alpha+1 and beta+1`.

Answer

To find the equation whose roots are \( \alpha + 1 \) and \( \beta + 1 \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 + 3x + 2 = 0 \), we can follow these steps:

### Step 1: Identify coefficients from the given quadratic equation
The given equation is \( 2x^2 + 3x + 2 = 0 \). Here, we identify:
- \( a = 2 \)
- \( b = 3 \)
- \( c = 2 \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) can be calculated using the formula:
\[ \alpha + \beta = -\frac{b}{a} \]
Substituting the values:
\[ \alpha + \beta = -\frac{3}{2} \]

### Step 3: Calculate the product of the roots
The product of the roots \( \alpha \beta \) can be calculated using the formula:
\[ \alpha \beta = \frac{c}{a} \]
Substituting the values:
\[ \alpha \beta = \frac{2}{2} = 1 \]

### Step 4: Calculate the new sum of the roots
Now, we need to find the sum of the new roots \( \alpha + 1 \) and \( \beta + 1 \):
\[ (\alpha + 1) + (\beta + 1) = (\alpha + \beta) + 2 \]
Substituting the value of \( \alpha + \beta \):
\[ = -\frac{3}{2} + 2 = -\frac{3}{2} + \frac{4}{2} = \frac{1}{2} \]

### Step 5: Calculate the new product of the roots
Next, we calculate the product of the new roots \( (\alpha + 1)(\beta + 1) \):
\[ (\alpha + 1)(\beta + 1) = \alpha \beta + \alpha + \beta + 1 \]
Substituting the values of \( \alpha \beta \) and \( \alpha + \beta \):
\[ = 1 + \left(-\frac{3}{2}\right) + 1 = 1 - \frac{3}{2} + 1 = 2 - \frac{3}{2} = \frac{4}{2} - \frac{3}{2} = \frac{1}{2} \]

### Step 6: Form the new quadratic equation
Now we can form the new quadratic equation using the sum and product of the new roots. The general form of a quadratic equation is:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - \left(\frac{1}{2}\right)x + \left(\frac{1}{2}\right) = 0 \]

### Step 7: Clear the fractions
To eliminate the fractions, we can multiply the entire equation by 2:
\[ 2x^2 - x + 1 = 0 \]

### Final Answer
Thus, the required quadratic equation whose roots are \( \alpha + 1 \) and \( \beta + 1 \) is:
\[ \boxed{2x^2 - x + 1 = 0} \]