Silverman exercise 3.1

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I try to do exercise 3.1 from graduate elliptic curve by Silverman. I find a discussion here

Silverman exercise 3.1 proving that two polynomials are relatively prime iff the discriminant is non-zero

But I cannot make sense of what discussed there. Is this supposed to be a proof oriented exercise ? But it seems the calculation is so complicated to do by hand. Also, there is an attempt to use resultant (?) which I guess somehow relates to discriminant.

In short, I am confused and not sure where to begin. What should be the associated Weierstrass form ? I mean there are two polynomails, does the associated form refers to $$E : y^2 = 4x^3 + b_2x^2 + 2b_4x + b_6 ?$$