To solve the equation \( 2x^{3} + 3^{2+x} = 891 \), let's first rewrite it clearly:

\[
2x^{3} + 3^{2+x} = 891
\]

Now, we need to solve for \( x \).

1. First, isolate \( 3^{2+x} \):

\[
3^{2+x} = 891 - 2x^{3}
\]

2. Now, we can try different integer values for \( x \) to see if we can find a solution.

Let's start with \( x = 5 \):

\[
2(5^{3}) + 3^{2+5} = 2 \cdot 125 + 3^{7}
\]
\[
= 250 + 2187 = 2437, \text{ (too high)}
\]

Now try \( x = 4 \):

\[
2(4^{3}) + 3^{2+4} = 2 \cdot 64 + 3^{6}
\]
\[
= 128 + 729 = 857, \text{ (close but too low)}
\]

Now try \( x = 4.5 \):

\[
2(4.5^{3}) + 3^{2+4.5} \approx 2 \cdot 91.125 + 3^{6.5} \approx 182.250 + 277.128 \approx 459.378, \text{ (still too low)}
\]

Now try \( x = 4.8 \):

\[
2(4.8^{3}) + 3^{2+4.8} \approx 2 \cdot 110.592 + 3^{6.8} \approx 221.184 + 637.75 \approx 858.934, \text{ (still low)}
\]

Now try \( x = 4.9 \):

\[
2(4.9^{3}) + 3^{2+4.9} \approx 2 \cdot 117.649 + 3^{6.9} \approx 235.298 + 874.6575 \approx 1110.05, \text{ (too high)}
\]

Through computation, I'll settle on \( x = 4.8 \) being close to the solution.

The precise value may be obtained by numerical methods or graphing. If you are interested in interpolation between values or using more advanced methods (like Newton-Raphson), that can lead to a more accurate solution.

In summary, the original equation can roughly be solved for \( x \) around the value of \( 4.8 \) to \( 4.9 \). For exact answers, numerical solving or graphing tools will be beneficial.