Solution to part I:

Factor the expression.

2xy + 2y + 3x = 42

2y(x+1) + 3(x+1) = 45

(x+1)(2y+3) = 45

  A  *   B   =  N+3

A  B   (x,y)  (x,y) for -A*-B

1  45  (0,21)   (-2,-24)

3  15  (2,6)    (-4,-9)

5  9   (4,3)    (-6,-6)

9  5   (8,1)    (-10,-4)

15 3   (14,0)   (-16,-3)

45 1   (44,-1)  (-46,-2)

 

Solution to part II:  

In general, considering the factors to be in ordered pairs, 

If 2-digit N replaces 42, then 

(x+1)(2y+3) = N+3

N is from 10 to 99 (2-digit number)

So N+3 is from 13 to 102

(the n in my code is really N+3)

N+3 [x,y]  N 

13 [0, 5] 10

14 [1, 2] 11

15 [0, 6] 12

15 [2, 1] 12

15 [4, 0] 12

17 [0, 7] 14

18 [1, 3] 15

18 [5, 0] 15

19 [0, 8] 16

20 [3, 1] 17

21 [0, 9] 18

21 [2, 2] 18

21 [6, 0] 18

22 [1, 4] 19

23 [0, 10] 20

24 [7, 0] 21

25 [0, 11] 22

25 [4, 1] 22

26 [1, 5] 23

27 [0, 12] 24

27 [2, 3] 24

27 [8, 0] 24

28 [3, 2] 25

29 [0, 13] 26

30 [1, 6] 27

30 [5, 1] 27

30 [9, 0] 27

31 [0, 14] 28

33 [0, 15] 30

33 [2, 4] 30

33 [10, 0] 30

34 [1, 7] 31

35 [0, 16] 32

35 [4, 2] 32

35 [6, 1] 32

36 [3, 3] 33

36 [11, 0] 33

37 [0, 17] 34

38 [1, 8] 35

39 [0, 18] 36

39 [2, 5] 36

39 [12, 0] 36

40 [7, 1] 37

41 [0, 19] 38

42 [1, 9] 39

42 [5, 2] 39

42 [13, 0] 39

43 [0, 20] 40

44 [3, 4] 41

45 [0, 21] 42

45 [2, 6] 42

45 [4, 3] 42

45 [8, 1] 42

45 [14, 0] 42

46 [1, 10] 43

47 [0, 22] 44

48 [15, 0] 45

49 [0, 23] 46

49 [6, 2] 46

50 [1, 11] 47

50 [9, 1] 47

51 [0, 24] 48

51 [2, 7] 48

51 [16, 0] 48

52 [3, 5] 49

53 [0, 25] 50

54 [1, 12] 51

54 [5, 3] 51

54 [17, 0] 51

55 [0, 26] 52

55 [4, 4] 52

55 [10, 1] 52

56 [7, 2] 53

57 [0, 27] 54

57 [2, 8] 54

57 [18, 0] 54

58 [1, 13] 55

59 [0, 28] 56

60 [3, 6] 57

60 [11, 1] 57

60 [19, 0] 57

61 [0, 29] 58

62 [1, 14] 59

63 [0, 30] 60

63 [2, 9] 60

63 [6, 3] 60

63 [8, 2] 60

63 [20, 0] 60

65 [0, 31] 62

65 [4, 5] 62

65 [12, 1] 62

66 [1, 15] 63

66 [5, 4] 63

66 [21, 0] 63

67 [0, 32] 64

68 [3, 7] 65

69 [0, 33] 66

69 [2, 10] 66

69 [22, 0] 66

70 [1, 16] 67

70 [9, 2] 67

70 [13, 1] 67

71 [0, 34] 68

72 [7, 3] 69

72 [23, 0] 69

73 [0, 35] 70

74 [1, 17] 71

75 [0, 36] 72

75 [2, 11] 72

75 [4, 6] 72

75 [14, 1] 72

75 [24, 0] 72

76 [3, 8] 73

77 [0, 37] 74

77 [6, 4] 74

77 [10, 2] 74

78 [1, 18] 75

78 [5, 5] 75

78 [25, 0] 75

79 [0, 38] 76

80 [15, 1] 77

81 [0, 39] 78

81 [2, 12] 78

81 [8, 3] 78

81 [26, 0] 78

82 [1, 19] 79

83 [0, 40] 80

84 [3, 9] 81

84 [11, 2] 81

84 [27, 0] 81

85 [0, 41] 82

85 [4, 7] 82

85 [16, 1] 82

86 [1, 20] 83

87 [0, 42] 84

87 [2, 13] 84

87 [28, 0] 84

88 [7, 4] 85

89 [0, 43] 86

90 [1, 21] 87

90 [5, 6] 87

90 [9, 3] 87

90 [17, 1] 87

90 [29, 0] 87

91 [0, 44] 88

91 [6, 5] 88

91 [12, 2] 88

92 [3, 10] 89

93 [0, 45] 90

93 [2, 14] 90

93 [30, 0] 90

94 [1, 22] 91

95 [0, 46] 92

95 [4, 8] 92

95 [18, 1] 92

96 [31, 0] 93

97 [0, 47] 94

98 [1, 23] 95

98 [13, 2] 95

99 [0, 48] 96

99 [2, 15] 96

99 [8, 4] 96

99 [10, 3] 96

99 [32, 0] 96

100 [3, 11] 97

100 [19, 1] 97

101 [0, 49] 98

102 [1, 24] 99

102 [5, 7] 99

102 [33, 0] 99

2-digit numbers that work

[10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99]

There are 87 of these, including 42

--------------

def factorPairs(n):

    ans = []

    for i in range(1,n+1):

        if (n/i)%1 == 0:

            ans.append([i,int(n/i)])

    return ans

nlist = []  

print('N+3 [x,y]  N ')      

for n in range(13,103):  # comment out one of these lines

# for n in range(45,46):  # comment out one of these lines

    for pair in factorPairs(n):

        y = (pair[1] - 3)/2

        if y%1 != 0:

            continue

        x = pair[0] - 1

        if x<0 or y<0:

            continue

        y = int(y)

        print(n, [x,y], 2*x*y+3*x+2*y )

        if n-3 not in nlist:

            nlist.append(n-3)

print(nlist )

print(len(nlist))