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The distributive property is a key concept in algebra that allows you to simplify expressions, especially when dealing with multiplication across terms within parentheses. The property states that for any numbers or expressions, such as variables, \[ a(b + c) = ab + ac \]This means you distribute the multiplier (in this case, \( a \)) to each term inside the parentheses.In our original exercise, the expression \( 2(a+1)(a-9) \), we utilize the distributive property twice. First, we distribute each term in \((a + 1)\) across every term in \((a - 9)\). This step expands the product of binomials as:

• \( a(a - 9) \) and
• \( 1(a - 9) \)

Resulting in the expanded expression:

• \( a^2 - 9a + a - 9 \)

Finally, we apply the distributive property one more time to distribute the constant \( 2 \) to each term in the simplified expression \( (a^2 - 8a - 9) \), giving the final result.

The FOIL method is specifically used for multiplying two binomials — that is, expressions with two terms. FOIL stands for First, Outer, Inner, Last, referring to the pairs of terms you multiply together to expand the expression.For the original problem \((a+1)(a-9)\), we apply the FOIL method step-by-step:

• First: Multiply the first terms in each binomial: \( a \times a = a^2 \)
• Outer: Multiply the outer terms: \( a \times -9 = -9a \)
• Inner: Multiply the inner terms: \( 1 \times a = a \)
• Last: Multiply the last terms in each binomial: \( 1 \times -9 = -9 \)

When you sum these products, you get the complete expanded expression:\[ a^2 - 9a + a - 9 \]This is exactly the starting point for combining like terms in the final simplification.

Combining like terms is a crucial step to simplify polynomial expressions, which involves summing coefficients of terms that share the same variables.In our exercise, after applying the FOIL method, we are left with:\[ a^2 - 9a + a - 9 \]To simplify, we identify and combine like terms.

• The terms \( -9a \) and \( a \) are like terms as they both contain \( a \) as the variable. Adding their coefficients: \( -9 + 1 = -8 \).

Therefore, our expression becomes:\[ a^2 - 8a - 9 \]which is now simplified by combining like terms.Finally, distributing the 2, we have:\[ 2a^2 - 16a - 18 \]This end result combines all terms correctly and is a tidy completion of the problem using these algebraic strategies.