How to find a modulus and argument of w that is a quotient of z1 and z2 such that z1 = 1 + root(3)i and z2 = 1+ i using modulus-argument form?

First of all, transform both z₁ and z₂ into modulus-argument form. To obtain that form, you are going to need a modulus (the length of the hypotenuse on the Argand diagram) and argument (the angle between opposite and adjacent). Modulus can be simply calculated using the Pythagoras's equation, so: modulus of z₁ is equal to the root of the sum of the squared lengths of both opposite and adjacent. In this case, modulus z₁ = 2 and modulus z₂ = root of 2. Getting angles is easy too. Simply calculate arctan of imaginary/real, but remember to draw the triangle on the diagram, as it is easy to make a mistake here, especially when dealing with angles in 3^rd and 4^th quadrants. The obtained angles should be pi/3 and pi/4.
Now onto the modulus-argument form. The general formula is: mod(z)[cos(arg) + isin(arg)], so try following that.
After doing so, to get mod(w) divide mod(z₁) and mod(z₂) as w is a quotient. The argument is arg(z₁) - arg(z₂).