In each of these questions two equations 1 and 2 are given.You have to solve both the equations and give answer
(A) if `a lt b`
(B) if `a gt b`
( C ) if relationship between a and b cannot been established
(D) if `a ge b`
(E) if `a le b`
(1) `a^2 + 5a + 6 = 0`
(2) `b^2 + 3b + 2 = 0`

Answer

To solve the given equations and determine the relationship between \( a \) and \( b \), we will follow these steps:

### Step 1: Solve the first equation \( a^2 + 5a + 6 = 0 \)

To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to \( 6 \) (the constant term) and add up to \( 5 \) (the coefficient of \( a \)).

The factors of \( 6 \) that satisfy this condition are \( 2 \) and \( 3 \). Thus, we can write:

\[ a^2 + 2a + 3a + 6 = 0 \]

Grouping the terms gives us:

\[ (a + 2)(a + 3) = 0 \]

Setting each factor to zero, we find:

\[ a + 2 = 0 \quad \Rightarrow \quad a = -2 \]
\[ a + 3 = 0 \quad \Rightarrow \quad a = -3 \]

Thus, the solutions for \( a \) are \( a = -2 \) and \( a = -3 \).

### Step 2: Solve the second equation \( b^2 + 3b + 2 = 0 \)

Similarly, we will factor this quadratic equation. We need to find two numbers that multiply to \( 2 \) and add up to \( 3 \).

The factors of \( 2 \) that satisfy this condition are \( 1 \) and \( 2 \). Thus, we can write:

\[ b^2 + 1b + 2b + 2 = 0 \]

Grouping the terms gives us:

\[ (b + 1)(b + 2) = 0 \]

Setting each factor to zero, we find:

\[ b + 1 = 0 \quad \Rightarrow \quad b = -1 \]
\[ b + 2 = 0 \quad \Rightarrow \quad b = -2 \]

Thus, the solutions for \( b \) are \( b = -1 \) and \( b = -2 \).

### Step 3: Compare the values of \( a \) and \( b \)

Now we have the solutions:
- For \( a \): \( -2 \) and \( -3 \)
- For \( b \): \( -1 \) and \( -2 \)

Now we will compare the values:
1. If \( a = -2 \) and \( b = -1 \), then \( a < b \).
2. If \( a = -3 \) and \( b = -2 \), then \( a < b \).
3. If \( a = -2 \) and \( b = -2 \), then \( a = b \).

### Conclusion

In all cases, the relationship we can establish is that \( a \leq b \).

### Final Answer

The correct answer is (E) if \( a \leq b \).

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