Directions (41-45):In each of these questions, two equations (I) and (II) are given. Solve the equations and mark the correct option.

I. 5x² + 11x + 2 = 0
II. 4y² + 13y + 3 = 0

The correct option is If x = y or no relation can be established between x and y.

To determine the relationship between $x$ and $y$, we need to solve both quadratic equations step-by-step.

Step 1: Solve Equation I for $x$

The given equation is:

$5x^2+11x+2=0$

We split the middle term $11x$ such that the product of the coefficients is $5\×2=10$ and their sum is $11$ (i.e., $10$ and $1$):

$5x^2+10x+1x+2=0$

$5x(x+2)+1(x+2)=0$

$(5x+1)(x+2)=0$

Setting each factor equal to zero gives:

$x=-2$ or $x=-\frac{1}{5}=-0.2$

Step 2: Solve Equation II for $y$

The given equation is:

$4y^2+13y+3=0$

We split the middle term $13y$ such that the product of the coefficients is $4\×3=12$ and their sum is $13$ (i.e., $12$ and $1$):

$4y^2+12y+1y+3=0$

$4y(y+3)+1(y+3)=0$

$(4y+1)(y+3)=0$

Setting each factor equal to zero gives:

$y=-3$ or $y=-\frac{1}{4}=-0.25$

Step 3: Compare the values of $x$ and $y$

Let's compare each value of $x$ with each value of $y$:

1. When $x=-0.2$ and $y=-0.25$: here $x>y$ because $-0.2$ is larger than $-0.25$.
2. When $x=-2$ and $y=-0.25$: here $x<y$ because $-2$ is smaller than $-0.25$.

Since we obtain both $x>y$ and $x<y$, no definite relationship can be established between $x$ and $y$.