1. **State the problem:** Solve the equation $$a(x - 1) - \frac{1 - 2x^2}{2} = (x - 4)(4 + x) + 3(1 + a)$$ for $a$ when $x = -\frac{7}{2}$.\n\n2. **Substitute $x = -\frac{7}{2}$ into the equation:**\n$$a\left(-\frac{7}{2} - 1\right) - \frac{1 - 2\left(-\frac{7}{2}\right)^2}{2} = \left(-\frac{7}{2} - 4\right)\left(4 - \frac{7}{2}\right) + 3(1 + a)$$\n\n3. **Simplify inside the parentheses:**\n$$a\left(-\frac{7}{2} - \frac{2}{2}\right) - \frac{1 - 2\left(\frac{49}{4}\right)}{2} = \left(-\frac{7}{2} - \frac{8}{2}\right)\left(\frac{8}{2} - \frac{7}{2}\right) + 3 + 3a$$\n$$a\left(-\frac{9}{2}\right) - \frac{1 - \frac{98}{4}}{2} = \left(-\frac{15}{2}\right)\left(\frac{1}{2}\right) + 3 + 3a$$\n\n4. **Simplify the fraction in the numerator:**\n$$1 - \frac{98}{4} = \frac{4}{4} - \frac{98}{4} = -\frac{94}{4} = -\frac{47}{2}$$\n\n5. **Substitute back:**\n$$a\left(-\frac{9}{2}\right) - \frac{-\frac{47}{2}}{2} = -\frac{15}{2} \times \frac{1}{2} + 3 + 3a$$\n\n6. **Simplify the left fraction:**\n$$-\frac{9}{2}a + \frac{47}{4} = -\frac{15}{4} + 3 + 3a$$\n\n7. **Simplify the right side constants:**\n$$-\frac{15}{4} + 3 = -\frac{15}{4} + \frac{12}{4} = -\frac{3}{4}$$\n\n8. **Rewrite the equation:**\n$$-\frac{9}{2}a + \frac{47}{4} = -\frac{3}{4} + 3a$$\n\n9. **Bring all terms involving $a$ to one side and constants to the other:**\n$$-\frac{9}{2}a - 3a = -\frac{3}{4} - \frac{47}{4}$$\n\n10. **Combine like terms:**\n$$-\frac{9}{2}a - \frac{6}{2}a = -\frac{50}{4}$$\n$$-\frac{15}{2}a = -\frac{50}{4}$$\n\n11. **Simplify the right side:**\n$$-\frac{15}{2}a = -\frac{25}{2}$$\n\n12. **Divide both sides by $-\frac{15}{2}$:**\n$$a = \frac{-\frac{25}{2}}{-\frac{15}{2}} = \frac{25}{2} \times \frac{2}{15} = \frac{25}{15} = \frac{5}{3}$$\n\n**Final answer:** $$a = \frac{5}{3}$$