Sure, let's go through the subtraction step-by-step to arrive at the solution.
We start with the given expression:
[tex]\[
\left(-5 b^2 - 8 b\right) - \left(-9 b^3 - 5 b^2 - 8 b\right)
\][/tex]
Step 1: Distribute the negative sign across the second polynomial expression to remove the parentheses.
[tex]\[
-5 b^2 - 8 b - (-9 b^3 - 5 b^2 - 8 b)
\][/tex]
This simplifies to:
[tex]\[
-5 b^2 - 8 b + 9 b^3 + 5 b^2 + 8 b
\][/tex]
Step 2: Combine like terms. Group together the terms with the same powers of [tex]\(b\)[/tex].
[tex]\[
(9 b^3) + (-5 b^2 + 5 b^2) + (-8 b + 8 b)
\][/tex]
Step 3: Simplify each group of like terms.
For [tex]\(b^3\)[/tex] terms:
[tex]\[
9 b^3
\][/tex]
For [tex]\(b^2\)[/tex] terms:
[tex]\[
-5 b^2 + 5 b^2 = 0
\][/tex]
For [tex]\(b\)[/tex] terms:
[tex]\[
-8 b + 8 b = 0
\][/tex]
So, the final result, after combining all the like terms, is:
[tex]\[
9 b^3
\][/tex]
Therefore, the polynomial in standard form is:
[tex]\[
9 b^3
\][/tex]
