For talking about things physically extending far down or far back in space, deep is almost always the appropriate word (a deep well, deep shelves, a deep cut). Profound is more elevated diction reserved for awe-invoking depths (profound depths of the sea). Both adjectives can also be used figuratively, of feeling and thought: we admire the deep thinkers and profound poets for their ability to penetrate beyond surface impressions and discover truths inaccessible to most of us. However, deep is more informal and used in this way can come off as a bit sophomoric (Wow, that’s a deep poem.) Profound suggests with more authority something’s intellectual significance (profound ideas or insights). In general usage, profound is most frequently used to describe change and its effects. In this context, profound implies far-reaching, transformative, significant, fundamental, or lasting (profound change, profound impact, profound influence).
The adjectives controversial and contentious both describe something that provokes differences of opinion or disagreements. A controversial issue, statement, or decision is one about which the public has strong and divergent opinions, often longstanding and much-discussed, and not likely to be reconciled any time soon. A contentious issue or topic is one that causes conflict or strife, whether in congress, in court, or at the dinner table. A contentious debate or election, or a contentious relationship with your supervisor, would be marked by hard-fought disagreements, as would a contentious divorce or contentious custody dispute. A controversial figure is a public figure who does or says controversial things. Contentious people are simply argumentative types.
To approve is to speak or think favorably of someone or something: to approve the policies of the administration. In some uses, approve signifies agreement or consent; in other uses, approve conveys a more formal confirmation or sanctioning, as when the Senate promptly approves a bill. The synonym endorse is most commonly used to talk about official and usually public displays of support. When someone endorses something, they approve, support, or sustain that thing. You may hear of a newspaper endorsing a candidate in a political campaign, or, of another politician endorsing, or putting their name behind, a candidate. The putting of one’s name to something takes a more literal turn in other uses of the word: to endorse a check is to designate oneself as payee by signing it, usually on the reverse side.