"Ella aún se aferraba a la esperanza de que encontraran a su perro con vida" or you can say "Ella aún tenía la esperanza de que encontraran a su perro con vida" In the first sentence the right word would be "encontraran".

"Ella aún se aferraba a la esperanza de que encontraran a su perro con vida" One letter.

it's a difficult one. This is an example of the "indefinite Preterite" and its different modes of use. The sentence is grammatically incorrect. The correct verb conjugation is "encontraran" (notice that is not the past form encontrarán -with the accent mark-) which is the subjunctive mode of the verb. If you say "encontraron", that means that They already found the dog, so it wouldn't make any sense in the sentence. "Encontraran" is the THIRD person of the PLURAL imperfect (Ellos) , and of course the sentence indicates that We don't know if the action (to find the dog alive) has been completed or not. In Spanish the subjunctive is applied directly to the verb, creating a new verb conjugation. The sentence is taking about something that already happened, so that's what creates a confusion, because you have to use the "imperfect preterite" form of the verb "Aferrar" into an "embedded sentence" so to speak (because in this case, someone is telling something that happened to other person in the past). If We were at the moment that the action happens the sentence would be: "Ella aún se aferra a la esperanza de que encuentren a su perro con vida", here we have to use the PRESENT subjunctive mode of the verb "Encontrar" (Encuentren) and the present form of the verb "Aferrar" (Aferra). Anyway, This grammar is as difficult as the many conjugations of one verb in Spanish. Practice is the way to get the notion of the use of this type of sentences and the many conjugations of The Spanish verbs, Spanish speakers don't know this rules by memorizing them, because it has a lot to do with the eloquence in the rhythm of the language, so you will find that there are situations when Spanish native speakers make mistakes with this sort of grammar.

I'll say: Ella aún se aferraba a la esperanza de que encuentren su perro con vida.

En mi opinión, podría haber varias vías para resolver gramaticalmente esa frase. La restricción para el tiempo verbal con el que corregir el "encontraron" viene del "aún se aferraba", que da impresión de un deseo/preocupación continuado. Situándolo en el presente, para resolver una situación que viene del pasado y que puede estar finalizando en ese momento, puedes decir "Ella aún se aferraba a la esperanza de que hubieran encontrado a su perro con vida" Un camino puede ser buscando fórmulas en subjuntivo, pero depende del momento en el que quieras colocar la acción y de cuánto lo quieras complicar: [...] de que podrían conseguir encontrar a su perro // [...] de que encontrasen a su perro. En cualquier caso, espero que el perro esté bien :-)