It's very hard to read. You may want to recreate the image.
IMO, answer is A

total 1400
42% - R (research)
W?? (W=women)
a)let women = x%. Thus men = (100-x)%
50% of x% + 36% of(100-x)% = 42%
we can solve for x
Sufficient
b) 288 men said R
Then we can find how many women said R but not total no. of women
Insufficient
IMO A

Of the 1400 teachers surveyed 42% said that they considered engaging in research an essential goal. How many of the teachers surveyed were women?

(1) In the survey 36% of the men and 50% of the women said that they considered engaging in research an essential goal.
(2) In the survey 288 men said that they considered engaging in research an essential goal.

This is a weighted average question: the men are being combined with the women to form a mixture (the 1400 teachers).
Weighted average DS questions are very common. Learn to recognize the pieces of the puzzle.
A weighted average question has 3 key facts:

Fact 1: The percentage contained within each element of the mixture.
Fact 2: The percentage contained within the final mixture.
Fact 3: The ratio of the elements within the final mixture.

If we are given Fact 1 and Fact 2, we can determine Fact 3.
If we are given Fact 1 and Fact 3, we can determine Fact 2.
No math is needed if we understand how the pieces of the puzzle fit together.

To determine how many of the 1400 teachers were women, we need to know Fact 3: the ratio of men to women.
We are given Fact 2: 42% of the 1400 teachers (the final mixture) considered engaging in research an essential goal.
Thus, we need Fact 1, the percentage contained within each element of the mixture.

Question rephrased: What percentage of the men and what percentage of the women (each element of the mixture) considered engaging in research an essential goal?

Statement 1: In the survey 36% of the men and 50% of the women said that they considered engaging in research an essential goal.
This is Fact 1: the percentage contained within each element of the mixture.
Sufficient.

Statement 2: In the survey 288 men said that they considered engaging in research an essential goal.
No way to determine the number of women.
Insufficient.

The correct answer is A.

Notice that no math is needed to determine the correct answer.

If this were a PS problem, we could use alligation to determine the ratio of men to women:

The proportion needed of each element within the mixture is equal to the distance between the percentage attributed to the other element within the mixture and the percentage attributed to the final mixture.

Proportion of men = Final mixture percentage - Women's percentage = 50-42 = 8.
Proportion of women = Men's percentage - Mixture percentage = 42-36 = 6.
Ratio of men to women = 8:6 = 4:3.

Since 4+3=7, 3/7 of the teachers are women.
(3/7)*1400 = 600.

A:
essential non essential
men 0.36x 0.64x x

women 0.5y 0.5y y

588 812 1400

So two equations and two unknowns. Sufficient

B.
essential non essential
men 288 x

women 300 y

588 812 1400

So here we cannot find Y. INsufficient

So IMO the answer is A

Sorry looks like the text area is removing spaces. I hope u can understand the matrix though..