Certainly! Let's expand the expression [tex]\((-f + 10g)(f - g)\)[/tex] and simplify it step-by-step.
### Step-by-Step Solution
1. Distributive Property (FOIL Method):
We will apply the distributive property, also known as the FOIL method (First, Outside, Inside, Last), to expand the product of the two binomials:
[tex]\[
(-f + 10g)(f - g)
\][/tex]
2. First Term:
The first terms in each binomial are [tex]\(-f\)[/tex] and [tex]\(f\)[/tex]. Multiply them together:
[tex]\[
(-f) \cdot (f) = -f^2
\][/tex]
3. Outside Term:
The outside terms are [tex]\(-f\)[/tex] and [tex]\(-g\)[/tex]. Multiply them:
[tex]\[
(-f) \cdot (-g) = fg
\][/tex]
4. Inside Term:
The inside terms are [tex]\(10g\)[/tex] and [tex]\(f\)[/tex]. Multiply them:
[tex]\[
10g \cdot (f) = 10fg
\][/tex]
5. Last Term:
The last terms in each binomial are [tex]\(10g\)[/tex] and [tex]\(-g\)[/tex]. Multiply them:
[tex]\[
10g \cdot (-g) = -10g^2
\][/tex]
6. Combine All Terms:
Now, combine all the terms from the First, Outside, Inside, and Last steps:
[tex]\[
-f^2 + fg + 10fg - 10g^2
\][/tex]
7. Simplify the Expression:
Combine like terms, specifically the [tex]\(fg\)[/tex] terms:
[tex]\[
fg + 10fg = 11fg
\][/tex]
So, the expression simplifies to:
[tex]\[
-f^2 + 11fg - 10g^2
\][/tex]
### Final Answer:
After expanding and simplifying the expression [tex]\((-f + 10g)(f - g)\)[/tex], the result is:
[tex]\[
-f^2 + 11fg - 10g^2
\][/tex]
