A group of students is investigating a non-linear elastic band. The magnitude of the restoring force \( F \) exerted by the band is modeled by the equation \( F = kx + bx^2 \), where \( x \) is the stretch distance, and \( k \) and \( b \) are positive constants. The students wish to experimentally determine \( k \) and \( b \), and then investigate whether mechanical energy is conserved when the band launches a cart of mass \( M \) along a horizontal track.

The students use a force sensor and a meterstick to measure the restoring force \( F \) at various stretch distances \( x \). Their data are shown in the table below. You may use the blank columns in the table to record any calculated values.

Stretch Distance \( x \) (\( \text{m} \))	Force \( F \) (\( \text{N} \))
\( 0.10 \)	\( 1.3 \)
\( 0.20 \)	\( 3.2 \)
\( 0.30 \)	\( 5.7 \)
\( 0.40 \)	\( 8.8 \)
\( 0.50 \)	\( 12.5 \)