Write the haplotypes of the family
Definition of Haplotype

The concept of haplotype has the default to be subject to several definiton that differs mostly by their complexity to be understood and by their specificity but different definitions also sometimes differ slightly in their meaning.

In simple terms the concept is pretty easy. A haplotype is a porte-manteau word of haploid genotype and it refers to half the genotype.

What the question is asking

The question is asking is Here is half of the genotype of the child, what do you expect the other half to be?. Note however, that for the last two markers, the first haplotype is not indicated so they are just asking for the genotypes here which makes the question slightly poorly phrased IMO.

The answer key

The answer concern the father, which is silly as the question is about the child. So the answer you stated definitely does not answer the question.

The answer

We have to look marker per marker to answer the question and then consider them all together to reconstruct the haplotype. In absence of information on the subject, we are going to assume that the markers are completely independent (unlinked, eventually on different chromosomes). We will also assume no mutation (probably negligible anyway) and we assume no crazy kind of inheritance such as some selfish elements. You will note also that information about the first child tells nothing about the second child.

Marker A

The first allele inherited necessarily comes from the mother and therefore, the second allele must comes from the father. As a consequence the second allele is either 3 or 4 with equal probability.

Marker B

The first allele inherited necessarily comes from the father and therefore, the second allele must comes from the mother. As a consequence the second allele is either 1 or 4 with equal probability.

Marker C

The four possible genotype are [5,7], [5,11], [11,7] or [11,11], that all occur with equal probability. This means that for Marker C, at one haplotype, the probability of being 5 or 7 is $\frac{1}{4}$ and the probability of being 11 is $\frac{1}{2}$.

Marker D

The four possible genotype are [4,7], [4,8], [7,7] or [7,8], that all occur with equal probability. This means that for Marker D, at one haplotype, the probability of being 4 or 8 is $\frac{1}{4}$ and the probability of being 7 is $\frac{1}{2}$.

Whole Haplotype

From the above, you can compute all possible haplotypes and calculate the most likely to occur. However, there are some ties. The most likely genotypes are (3,1,11,7), (3,4,11,7), (4,1,11,7) and (4,4,11,7) that could occur with probability $\left(\frac{1}{2}\right)^4 = \frac{1}{16}$.