Ans: (b) 32 : 65

Explanation:

\displaystyle \frac{a}{b}=\displaystyle\frac{2}{3}
\begin{aligned} \frac{3a^3+4b^2a}{5a^2b+5b^3}&=\frac{3(\frac{a}{b})^3 +4(\frac{a}{b})}{5(\frac{a}{b})^2 +5}\\[1.5em] &=\frac{3(\frac{2}{3})^3 +4(\frac{2}{3})}{5(\frac{2}{3})^2 +5}\\[1.5em] &=\frac{3\times \frac{8}{27} +\frac{8}{3}}{5\times \frac{4}{9} +5}\\[1.5em] &=\frac{\frac{8}{9} +\frac{8}{3}}{\frac{20}{9} +5}\\[1.5em] &=\frac{8+24}{20+45}\\[1.5em] &=\frac{32}{65} \end{aligned}

Aliter:

If you are only interested in answer try solving in the following way:

\displaystyle \frac{a}{b}=\displaystyle\frac{2}{3} \Rightarrow put a = 2, b = 3 in the expression

\begin{aligned} \frac{3a^3+4b^2a}{5a^2b+5b^3}&=\frac{3\times 8 + 4\times 9\times 2}{5 \times 4\times 3 + 5\times 27}\\[1.6em] &=\frac{8+24}{20+45}\quad \text{Dividing both numerator and denominator by 3}\\[1.6em] &=\frac{32}{65} \end{aligned}

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