To solve the equation \(2|2x + 3| = 10\), we first divide both sides by 2:
\[ |2x + 3| = 5 \]
Now, we will consider the two cases for the absolute value.
Case 1: \(2x + 3 = 5\)
To solve for \(x\):
\[ 2x + 3 = 5 \] \[ 2x = 5 - 3 \] \[ 2x = 2 \] \[ x = 1 \]
Case 2: \(2x + 3 = -5\)
To solve for \(x\) again:
\[ 2x + 3 = -5 \] \[ 2x = -5 - 3 \] \[ 2x = -8 \] \[ x = -4 \]
So the solutions to the equation \(2|2x + 3| = 10\) are:
\[ x = 1 \quad \text{and} \quad x = -4 \]
Thus, the final answers are:
\[ x = 1 \] \[ x = -4 \]