In the given questions, two quantities are given, one as Quantity I and another as Quantity II. You have to determine relationship between two quantities and choose the appropriate option.
`(x^2/5)+x+(4/5)=0`
`3y^2+4y+1=0`
Quantity I. x Quantity II. y

Options

A. If Quantity I `gt` Quantity
B. If Quantity I`ge` Quantity II
C. If Quantity I`gt`Quantity II
D. If Quantity I`le`Quantity II

Answer

To solve the given problem, we need to analyze the two equations provided and find the values of \( x \) and \( y \).

### Step 1: Solve for \( x \)

We start with the equation:

\[ \frac{x^2}{5} + x + \frac{4}{5} = 0 \]

To eliminate the fraction, we can multiply the entire equation by 5:

\[ 5 \left(\frac{x^2}{5}\right) + 5x + 5 \left(\frac{4}{5}\right) = 0 \]

This simplifies to:

\[ x^2 + 5x + 4 = 0 \]

### Step 2: Factor the quadratic equation for \( x \)

Next, we need to factor the quadratic equation \( x^2 + 5x + 4 = 0 \). We look for two numbers that multiply to 4 (the constant term) and add to 5 (the coefficient of \( x \)). The numbers 4 and 1 satisfy this:

\[ (x + 4)(x + 1) = 0 \]

### Step 3: Find the values of \( x \)

Setting each factor to zero gives us:

\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

Thus, the possible values for \( x \) are \( -4 \) and \( -1 \).

### Step 4: Solve for \( y \)

Now we consider the second equation:

\[ 3y^2 + 4y + 1 = 0 \]

### Step 5: Factor the quadratic equation for \( y \)

To factor \( 3y^2 + 4y + 1 = 0 \), we look for two numbers that multiply to \( 3 \times 1 = 3 \) and add to 4. The numbers 3 and 1 work:

\[ (3y + 1)(y + 1) = 0 \]

### Step 6: Find the values of \( y \)

Setting each factor to zero gives us:

\[ 3y + 1 = 0 \quad \Rightarrow \quad y = -\frac{1}{3} \]
\[ y + 1 = 0 \quad \Rightarrow \quad y = -1 \]

Thus, the possible values for \( y \) are \( -\frac{1}{3} \) and \( -1 \).

### Step 7: Compare the values of \( x \) and \( y \)

Now we compare the values of \( x \) and \( y \):

1. If \( x = -4 \):
- \( y = -1 \): \( -4 < -1 \)
- \( y = -\frac{1}{3} \): \( -4 < -\frac{1}{3} \)

2. If \( x = -1 \):
- \( y = -1 \): \( -1 = -1 \)
- \( y = -\frac{1}{3} \): \( -1 < -\frac{1}{3} \)

### Conclusion

From the comparisons, we can conclude:
- In all cases, \( x \) is less than or equal to \( y \).

Thus, the relationship between Quantity I (x) and Quantity II (y) is:

\[ \text{Quantity I} \leq \text{Quantity II} \]

### Final Answer

The correct option is **4**: Quantity I is less than or equal to Quantity II.