To find the average speed of the particle between \( t = 2 \) seconds and \( t = 5 \) seconds, we use the formula for average speed:

\(\text{Average Speed} = \frac{\Delta x}{\Delta t}\)

Calculate \( x \) at \( t = 2 \) seconds}

Using the equation \( x = 10 + 12t^2 \):

\(x(2) = 10 + 12(2^2) = 10 + 12(4) = 10 + 48 = 58 \, \text{m}\)

Calculate \( x \) at \( t = 5 \) seconds

\(x(5) = 10 + 12(5^2) = 10 + 12(25) = 10 + 300 = 310 \, \text{m}\)

Calculate \( \Delta x \) and \( \Delta t \)

\(\Delta x = x(5) - x(2) = 310 - 58 = 252 \, \text{m}\)
\(\Delta t = 5 - 2 = 3 \, \text{s}\)

Calculate the average speed

\(\text{Average Speed} = \frac{252 \, \text{m}}{3 \, \text{s}} = 84 \, \text{m/s}\).