Expand each of the following products. Your answers should be in a form similar to $x^{2}+6 x+5$. a. $(x-5)(x+4)=$ Preview b. $(x-14)(x-27)=$ Preview c. $(x-7)(x+7)=$ Preview d. $(x-y)^{2}=$ Preview
Solution
Step 1 ：The question is asking to expand the given expressions. This can be done by applying the distributive property of multiplication over addition, also known as the FOIL method (First, Outer, Inner, Last).
Step 2 ：For example, for the expression \((x-5)(x+4)\), we can apply the FOIL method as follows: First: Multiply the first terms in each binomial, \(x * x = x^2\). Outer: Multiply the outer terms in the product, \(x * 4 = 4x\). Inner: Multiply the inner terms in the product, \(-5 * x = -5x\). Last: Multiply the last terms in each binomial, \(-5 * 4 = -20\). Then, we add these results together to get the expanded form of the expression.
Step 3 ：Let's apply this method to the given expressions.
Step 4 ：For \((x-5)(x+4)\), the expanded form is \(x^{2} - x - 20\).
Step 5 ：For \((x-14)(x-27)\), the expanded form is \(x^{2} - 41x + 378\).
Step 6 ：For \((x-7)(x+7)\), the expanded form is \(x^{2} - 49\).
Step 7 ：For \((x-y)^{2}\), the expanded form is \(x^{2} - 2xy + y^{2}\).
Step 8 ：Final Answer: a. \((x-5)(x+4) = \boxed{x^{2} - x - 20}\) b. \((x-14)(x-27) = \boxed{x^{2} - 41x + 378}\) c. \((x-7)(x+7) = \boxed{x^{2} - 49}\) d. \((x-y)^{2} = \boxed{x^{2} - 2xy + y^{2}}\)
