Check whether the following is a quadratic equation:
$x^{2}-2x=(-2)(3-x)$

Represent the following situation in the form of a quadratic equation:
The product of two consecutive positive integers is $306$. We need to find the integers.

Check whether the following is a quadratic equation:
$(x-2)(x+1)=(x-1)(x+3)$

Check whether the following is a quadratic equation:
$(2x - 1)(x - 3) = (x + 5)(x - 1)$

$A$ prayer hall has a carpet area of $300 \ m^2$ with its length one metre more than twice its breadth. Find the dimensions of the prayer hall.

Represent the following situation mathematically:
John and Jivanti together have $45$ marbles. Both of them lost $5$ marbles each,and the product of the number of marbles they now have is $124$. We would like to find out how many marbles they had to start with.