Simplifying the Expression: (a  2)(a + 4)  (a + 1)^2
This article will guide you through simplifying the algebraic expression (a  2)(a + 4)  (a + 1)^2. We will use the distributive property and the FOIL method to expand the terms and then combine like terms to reach a simplified expression.
Expanding the Terms
First, we need to expand the terms using the distributive property:

(a  2)(a + 4): We multiply each term in the first set of parentheses by each term in the second set:
 a * a = a²
 a * 4 = 4a
 2 * a = 2a
 2 * 4 = 8
 (a  2)(a + 4) = a² + 4a  2a  8

(a + 1)²: This term represents the square of a binomial, so we can use the FOIL method (First, Outer, Inner, Last):
 First: a * a = a²
 Outer: a * 1 = a
 Inner: 1 * a = a
 Last: 1 * 1 = 1
 (a + 1)² = a² + a + a + 1
Combining Like Terms
Now we can substitute the expanded expressions back into the original equation and simplify:
(a  2)(a + 4)  (a + 1)² = (a² + 4a  2a  8)  (a² + a + a + 1)
Combine like terms within each set of parentheses:
(a² + 2a  8)  (a² + 2a + 1)
Finally, subtract the terms within the second set of parentheses from the first set:
a² + 2a  8  a²  2a  1
= 9
Conclusion
Therefore, the simplified form of the expression (a  2)(a + 4)  (a + 1)² is 9. This means the expression will always equal 9 regardless of the value of 'a'.