Sure, let's solve the given expression step by step:
Given expression:
[tex]\[
\frac{24a^3 - 8a^2 + 4a}{-4a}, \quad a \neq 0
\][/tex]
Step 1: Separate each term in the numerator and divide by the denominator:
[tex]\[
\frac{24a^3}{-4a} - \frac{8a^2}{-4a} + \frac{4a}{-4a}
\][/tex]
Step 2: Simplify each fraction individually:
[tex]\[
\frac{24a^3}{-4a} = 24a^{3-1} \cdot \frac{1}{-4} = \frac{24a^2}{-4} = -6a^2
\][/tex]
[tex]\[
\frac{8a^2}{-4a} = 8a^{2-1} \cdot \frac{1}{-4} = \frac{8a}{-4} = -2a
\][/tex]
[tex]\[
\frac{4a}{-4a} = \frac{4 \cdot a}{-4 \cdot a} = -1
\][/tex]
Step 3: Combine the simplified terms:
[tex]\[
-6a^2 + (-2a) + (-1)
\][/tex]
Step 4: Simplify the signs in the expression:
[tex]\[
-6a^2 - 2a - 1
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
-6a^2 - 2a - 1
\][/tex]
