First, express \(\frac{4x^2 + 5x + 3}{2x^2 + 5x + 2}\) in partial fractions:
Assume \(\frac{4x^2 + 5x + 3}{2x^2 + 5x + 2} = A + \frac{B}{2x+1} + \frac{C}{x+2}\).
Equating coefficients, solve for \(A, B,\) and \(C\):
\(A = 2, \; B = 1, \; C = -3\).
Thus, \(\frac{4x^2 + 5x + 3}{2x^2 + 5x + 2} = 2 + \frac{1}{2x+1} - \frac{3}{x+2}\).
Now integrate each term separately:
\(\int_0^4 \left( 2 + \frac{1}{2x+1} - \frac{3}{x+2} \right) \, dx\).
\(= \left[ 2x + \frac{1}{2} \ln |2x+1| - 3 \ln |x+2| \right]_0^4\).
Calculate the definite integral:
\(= \left( 8 + \frac{1}{2} \ln 9 - 3 \ln 6 \right) - \left( 0 + \frac{1}{2} \ln 1 - 3 \ln 2 \right)\).
\(= 8 + \frac{1}{2} \ln 9 - 3 \ln 6 + 3 \ln 2\).
Simplify the expression:
\(= 8 + \frac{1}{2} \ln 9 - 3 \ln 3 - 3 \ln 2 + 3 \ln 2\).
\(= 8 - \ln 9\).
