Answer :

If [tex]\(-\frac{3}{2}\)[/tex] is a zero of the polynomial [tex]\(a x^3 - x^2 + x + 4\)[/tex], it means that substituting [tex]\(x = -\frac{3}{2}\)[/tex] into the polynomial should result in zero. This can be expressed mathematically as:

[tex]\[a \left(-\frac{3}{2}\right)^3 - \left(-\frac{3}{2}\right)^2 + \left(-\frac{3}{2}\right) + 4 = 0\][/tex]

Let's substitute [tex]\(x = -\frac{3}{2}\)[/tex] step-by-step:

1. Calculate [tex]\(\left(-\frac{3}{2}\right)^3\)[/tex]:
[tex]\[ \left(-\frac{3}{2}\right)^3 = -\frac{27}{8} \][/tex]

2. Calculate [tex]\(\left(-\frac{3}{2}\right)^2\)[/tex]:
[tex]\[ \left(-\frac{3}{2}\right)^2 = \frac{9}{4} \][/tex]

3. Substitute these into the polynomial and simplify:
[tex]\[ a \left(-\frac{27}{8}\right) - \frac{9}{4} - \frac{3}{2} + 4 = 0 \][/tex]

4. Combine like terms:
[tex]\[ -\frac{27a}{8} - \frac{9}{4} - \frac{3}{2} + 4 = 0 \][/tex]

To simplify the equation, let's find a common denominator for the fractions:
[tex]\[ -\frac{27a}{8} - \frac{18}{8} - \frac{12}{8} + \frac{32}{8} = 0 \][/tex]

Combining the fractions:
[tex]\[ -\frac{27a}{8} - \frac{18 + 12 - 32}{8} = 0 \][/tex]

Simplify inside the parentheses:
[tex]\[ -\frac{27a}{8} + \frac{2}{8} = 0 \][/tex]

Simplify further:
[tex]\[ -\frac{27a + 2}{8} = 0 \][/tex]

Since the fraction equals zero, the numerator must be zero:
[tex]\[ -27a + 2 = 0 \][/tex]

Solving for [tex]\(a\)[/tex]:
[tex]\[ -27a = -2 \][/tex]
[tex]\[ a = \frac{2}{27} \][/tex]

Therefore, the value of [tex]\(a\)[/tex] is [tex]\(\boxed{\frac{2}{27}}\)[/tex].