In an effort to improve the quality of patient care, Dr. Lydia Temoscho is directing one of several clinical research projects that
seek to determine the extent that psychological counseling helps to supplement the medical treatment of serious disease.
Option Elimination -
That introduces a relative clause modifying "projects," so we need plural seek.
Just for clarity
What is the difference between "help in ING" and "help to"/The right usage for extent is - the extent of or the extent to which
"Help in" is used when we want to say that the assistance contributes to achieving a specific goal. E.g., The new software will help in improving productivity. Here, "the new software" contributes to achieving a specific goal which is improving productivity.
"Help to" is used when we say that the assistance enables someone to perform a specific task. E.g., The tutor helps to develop student's problem-solving skills. Here, the assistance provided by the teacher enables the students to develop problem-solving skills)
The right usage for extent is - the extent of or the extent to which
The "extent of" is used when discussing a degree of something without mentioning a criteria or a specific condition. E.g., The report discussed the extent of environmental damage caused by the oil spill. Here, the report examined the degree or magnitude of the environmental damage.
The "extent to which" is used to discuss the degree to which something is true in relation to a specific condition or criterion. It is typically followed by a clause or phrase that provides more information about the specific condition or criterion being evaluated.
E.g., The study examined the extent to which technology influences students' learning outcomes. Here, the study analyzed the degree or level to which technology impacts learning outcomes.
(A) seek to determine the extent that psychological counseling helps to supplement - the "extent to which" is preferred here as we have a criteria here at the end regarding how helpful psychological counseling is in supplementing the medical treatment of serious disease.
(B) seek to determine how helpful psychological counseling is in supplementing - In our case; we want to say that the assistance provided in terms of "psychological counseling" is how helpful it is to achieve a specific goal, which is the medical treatment of a serious disease. Also, the plural "seek" agrees with the plural "projects."
(C) seeks to determine how helpful psychological counseling is to supplement - SV issues.
(D) seeks to determine to what extent psychological counseling is a help in supplementing - SV issues.
(E) seeks to determine the extent that psychological counseling is helpful in supplementing - SV Issues