Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
Give the main organic product of each reaction, and name the rule or effect that decides it.
An alkene adds HX by Markovnikov's rule: the more stable carbocation forms, so the halide goes to the carbon with more alkyl groups. A carbocation can rearrange to a more stable one by a hydride or methyl shift. With a peroxide, HBr adds by a radical chain to give the anti-Markovnikov product, because the more stable radical forms.
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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
(a) The hydrogen goes to the carbon with one hydrogen, which leaves the tertiary cation. Bromide adds there: 2-bromo-2-methylbutane, . Markovnikov's rule.
(b) With a peroxide the bromine atom adds first, to the carbon with one hydrogen atom, so the more stable tertiary radical forms on the other carbon. The product is 2-bromo-3-methylbutane, . Peroxide effect, anti-Markovnikov.
(c) The cation forms on carbon 2: 2-iodobutane, . Markovnikov's rule. Hydrogen iodide shows no peroxide effect.
(d) The first cation, secondary, forms on carbon 2. A hydride shift from carbon 3 makes a tertiary cation. Bromide adds to it: 2-bromo-2-methylbutane is the main product. A little 2-bromo-3-methylbutane forms without the shift. Rearrangement.
Answer: (a) and (d) give 2-bromo-2-methylbutane, (b) gives 2-bromo-3-methylbutane, and (c) gives 2-iodobutane.
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