If the product of the roots of the equation `(a+1)x^2+(2a+3)x+(3a+4)=0i s2,` then find the sum roots.

Explanation

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Identify the equation and the condition
The given quadratic equation is:
\[ (a + 1)x^2 + (2a + 3)x + (3a + 4) = 0 \]
We know that the product of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[ \text{Product of roots} = \frac{c}{a} \]
In this case, \(c = 3a + 4\) and \(a = a + 1\).

### Step 2: Set up the equation for the product of the roots
According to the problem, the product of the roots is equal to 2:
\[ \frac{3a + 4}{a + 1} = 2 \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3a + 4 = 2(a + 1) \]

### Step 4: Expand and simplify the equation
Expanding the right side:
\[ 3a + 4 = 2a + 2 \]
Now, rearranging the equation:
\[ 3a - 2a = 2 - 4 \]
This simplifies to:
\[ a = -2 \]

### Step 5: Find the sum of the roots
The sum of the roots of a quadratic equation is given by:
\[ \text{Sum of roots} = -\frac{b}{a} \]
Here, \(b = 2a + 3\) and \(a = a + 1\). Substituting \(a = -2\):
\[ b = 2(-2) + 3 = -4 + 3 = -1 \]
And:
\[ a = -2 + 1 = -1 \]
Now substituting these values into the formula for the sum of the roots:
\[ \text{Sum of roots} = -\frac{-1}{-1} = -1 \]

### Final Answer
The sum of the roots is:
\[ \text{Sum of roots} = -1 \]

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