Can you please explain this step?

Do you understand it up to here  ?

\((x-2)(x^2+bx+c)=x^3-4x^2-11x+30\)

The bit in the box is just equating coefficients.

On the LHS the x^2 terms are      \(bx^2\;\; and \;\;-2x^2\)  

and on the right the x^2 term is      \( -4x^2\)

so

\(bx^2\;\;\ +\;\; -2x^2\;=\;-4x^2\\ (b-2)x^2=-4x^2\\ b-2=-4\\ b=-2\\ \)

Is that all you need help with ?

So if I'm understanding this correctly, bx^2 - 2x^2 = bx^2 (as per the general polynomial expression). Therefore as these 2 equations can be equated, you can match up the bx^2 values of both sides to equal each other?

This can also be done for any other variable too, and as c = -2c on the LHS and +30 on the RHS, c = -15.

Yes this is correct :)