Learn Spanish (Latin America): Ellos explicaron que expandir el negocio tendría un riesgo. - They explained that expanding the business would have a risk.

Because the main verb explicaron (they explained) is in the past. When someone in the past talked about their future (e.g., "Expanding the business will have a risk"), Spanish uses reported speech rules. Just like English changes "will" to "would," Spanish changes the future tense (tendrá) to the conditional (tendría).

How is tendría formed? I thought conditional verbs just added endings to the full infinitive.

Usually, they do! But tener is one of a few irregular verbs in the conditional (and future) tense. Instead of keeping the full infinitive, it uses the irregular stem tendr-, which gives us tendría instead of the incorrect form tenería.

Why does the Spanish sentence use the infinitive expandir when the English translation uses the -ing word "expanding"?

In English, we use the -ing form (a gerund) when a verb acts as a noun, like being the subject of the phrase "expanding the business". In Spanish, you can never use the -ndo form for this. You must use the infinitive (expandir) whenever a verb acts as a noun.

Is it common to say tener un riesgo? I thought "to take a risk" was correr un riesgo.