use the Newton-Raphson method.
Analysis of the graph of f(x) shows that it has two turning points - one at x = 0 and the other at x = -4/7.
Both f(0) and f(-4/7) are negative, i.e. the graph (of f(x)) between the turning points lies below the x-axis, therefore there is only one (real) root.
f(0) = -2 and f(1) = 11 - i.e. we have a change in sign (of f(x)) between x = 0 and x = 1, therefore the graph of f(x) must cross the x-axis between x = 0 and x = 1. So our root will lie in the intervsl [0 .. 1].
So take 1st estimate for the root as x1 = 0.5.
Newton-Raphson method
The iteration formula for the above is,
x_(k+1) = x_k - f(x_k)/f'(x_k)
Three iterations were required to give a root value of x = 0.464885
Answer: root = 0.464885