1. **State the problem:** Given two parallel lines $p \parallel n$ cut by a vertical and a diagonal transversal, find the value of $x$ using the given angles: $(5x + 5)^\circ$, $4y^\circ$, $2x^\circ$, and $88^\circ$. 2. **Identify relationships:** Since $p \parallel n$, corresponding and alternate interior angles are equal. 3. **Use the vertical transversal angles:** The angles $2x^\circ$ and $88^\circ$ are on a straight line, so they are supplementary. 4. **Write the supplementary angle equation:** $$2x + 88 = 180$$ 5. **Solve for $x$:** $$2x = 180 - 88$$ $$2x = 92$$ $$x = \frac{92}{2}$$ $$x = 46$$ 6. **Check with other angles:** The angle $(5x + 5)^\circ$ should be consistent with $x=46$. 7. **Calculate $(5x + 5)$:** $$5(46) + 5 = 230 + 5 = 235^\circ$$ Since $235^\circ$ is not a valid angle here, the key step is the supplementary angle relation between $2x$ and $88$ which gives $x=46$. **Final answer:** $$\boxed{46}$$