To tackle the problem of simplifying the expression [tex]\( 5 - 5(2 + x)(3 - x) \)[/tex], we will break it down step by step.
1. Distribute inside the parentheses: First, we need to expand the product [tex]\((2 + x)(3 - x)\)[/tex].
2. Use the distributive property (FOIL method):
[tex]\[
(2 + x)(3 - x) = 2 \cdot 3 + 2 \cdot (-x) + x \cdot 3 + x \cdot (-x)
\][/tex]
Breaking it down:
[tex]\[
= 6 - 2x + 3x - x^2
\][/tex]
3. Combine like terms within the expanded expression:
[tex]\[
6 - 2x + 3x - x^2 = 6 + x - x^2
\][/tex]
4. Consider the expression within the original problem: Now replace the expanded form back into the original expression.
[tex]\[
5 - 5(6 + x - x^2)
\][/tex]
5. Distribute [tex]\(-5\)[/tex] through the parentheses: Multiply each term inside the parentheses by [tex]\(-5\)[/tex]:
[tex]\[
5 - (5 \cdot 6 + 5 \cdot x - 5 \cdot x^2) = 5 - (30 + 5x - 5x^2)
\][/tex]
6. Simplify the result: Expand the subtraction:
[tex]\[
5 - 30 - 5x + 5x^2
\][/tex]
7. Combine the constants and terms:
[tex]\[
5x^2 - 5x - 25
\][/tex]
Thus, the simplified form of the expression [tex]\(5 - 5(2 + x)(3 - x)\)[/tex] is:
[tex]\[
5x^2 - 5x - 25
\][/tex]
This is the final simplified expression.
