Let's factor the expression [tex]\(-4d^3 + 28d^2 - 4d\)[/tex] step-by-step.
1. Identify the common factor:
First, observe that each term in the expression [tex]\(-4d^3 + 28d^2 - 4d\)[/tex] contains a factor of [tex]\(-4d\)[/tex].
2. Factor out the greatest common factor (GCF):
We begin by factoring out [tex]\(-4d\)[/tex] from each term in the expression:
[tex]\[
-4d^3 + 28d^2 - 4d = -4d(d^2 - 7d + 1)
\][/tex]
3. Verify the factorization (optional, but good practice):
To verify that this factorization is correct, you can distribute [tex]\(-4d\)[/tex] back through the factored expression:
[tex]\[
-4d(d^2 - 7d + 1) = -4d \cdot d^2 + (-4d) \cdot (-7d) + (-4d) \cdot 1
\][/tex]
Simplifying each term, this becomes:
[tex]\[
-4d^3 + 28d^2 - 4d
\][/tex]
Since this matches the original expression, the factorization is correct.
So, the fully factored form of the expression [tex]\(-4d^3 + 28d^2 - 4d\)[/tex] is:
[tex]\[
-4d(d^2 - 7d + 1)
\][/tex]
