To solve the expression [tex]\(\left(2 x^{m+1}+x^{m+2}-x^m\right)\left(x^{m+1}-2 x^{m+1}\right)\)[/tex], we need to break it down step by step. Let’s go through the multiplication and simplification process in detail:
First, simplify the second term inside the parentheses:
[tex]\[
x^{m+1} - 2 x^{m+1}
\][/tex]
Notice that both terms have a common factor of [tex]\(x^{m+1}\)[/tex], so we can factor it out:
[tex]\[
x^{m+1} - 2 x^{m+1} = (1 - 2)x^{m+1} = -x^{m+1}
\][/tex]
Now we substitute back into the original expression:
[tex]\[
\left(2 x^{m+1} + x^{m+2} - x^m\right) \left(-x^{m+1}\right)
\][/tex]
Next, distribute [tex]\(-x^{m+1}\)[/tex] across each term inside the first set of parentheses:
[tex]\[
(2 x^{m+1})(-x^{m+1}) + (x^{m+2})(-x^{m+1}) + (-x^m)(-x^{m+1})
\][/tex]
Now, let’s multiply each term:
1. [tex]\((2 x^{m+1})(-x^{m+1}) = -2 x^{(m+1) + (m+1)} = -2 x^{2m+2}\)[/tex]
2. [tex]\((x^{m+2})(-x^{m+1}) = -x^{(m+2) + (m+1)} = -x^{2m+3}\)[/tex]
3. [tex]\((-x^m)(-x^{m+1}) = x^{m + (m+1)} = x^{2m+1}\)[/tex]
So, combining these results, we get:
[tex]\[
-2 x^{2m+2} - x^{2m+3} + x^{2m+1}
\][/tex]
Thus, the final simplified expression is:
[tex]\[
x^{m + 1} \left(x^m - 2 x^{m + 1} - x^{m + 2}\right)
\][/tex]
This is the complete, detailed solution to the given expression.
