To factor the expression \(2xy + 3 + 2y + 3x\), we can first rearrange the terms:
\[ 2xy + 3x + 2y + 3 \]
Next, we can group the terms:
\[ (2xy + 3x) + (2y + 3) \]
Now, we can factor out common factors from each group:
From the first group, \(2xy + 3x\), we can factor out \(x\):
\[ x(2y + 3) \]
From the second group, \(2y + 3\), we leave it as is since there are no common factors to factor out.
Now, putting it together:
\[ x(2y + 3) + 1(2y + 3) \]
Notice that \(2y + 3\) is common in both terms, so we can factor it out:
\[ (2y + 3)(x + 1) \]
Thus, the factorized form of the expression \(2xy + 3 + 2y + 3x\) is:
\[ (2y + 3)(x + 1) \]