1. We are given a $3\times 3$ matrix $A$ such that $\det(A)=4.$

2. First consider the matrix $2A$.

For an $n\times n$ matrix, if every entry is multiplied by a scalar $k$, then the determinant gets multiplied by $k^n$: $\det(kA)=k^n\det(A).$ Here, $n=3$ and $k=2$, so $\det(2A)=2^3\det(A)=8\cdot 4=32.$

3. Now matrix $B$ is obtained from $2A$ by the row operation $R_2 \to 2R_2+5R_3.$

Let us split this into standard determinant effects:

• Multiplying a row by $2$ multiplies the determinant by $2$.
• Adding $5R_3$ to $R_2$ does not change the determinant.

So the net effect of the operation $R_2 \to 2R_2+5R_3$ is to multiply the determinant by $2$. Hence, $\det(B)=2\det(2A)=2\cdot 32=64.$

4. Therefore, the correct option is $\boxed{64}.$

5. Comparing with the stored correct answer:

• Derived answer: $64$
• Stored correct answer: A = $64$

So they agree.