Use PEMDAS (Parentheses, Exponents, Multipy/Divide, Add/Subtract):

2 + (3 + 4) ÷ 2 x 5 - 3²
P: 2 + (7) ÷ 2 x 5 - 3²
E: 2 + 7 ÷ 2 x 5 - 9
MD: 2 + \( \frac{7}{2} \) x 5 - 9
MD: 2 + \( \frac{35}{2} \) - 9
AS: \( \frac{4}{2} \) + \( \frac{35}{2} \) - 9
AS: \( \frac{39}{2} \) - 9
AS: \( \frac{39 - 18}{2} \)
\( \frac{21}{2} \)
10\(\frac{1}{2}\)

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 16 are [16, 32, 48, 64, 80, 96]. The first few multiples they share are [16, 32, 48, 64, 80] making 16 the smallest multiple 8 and 16 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{2 x 2}{8 x 2} \) + \( \frac{7 x 1}{16 x 1} \)

\( \frac{4}{16} \) + \( \frac{7}{16} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{4 + 7}{16} \) = \( \frac{11}{16} \) = \(\frac{11}{16}\)

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for distance:

distance = \( \text{speed} \times \text{time} \)
distance = \( 15mph \times 2h \)
30 miles

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{3!}{4!} \)
\( \frac{3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} \)
\( \frac{1}{4} \)
\( \frac{1}{4} \)

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).