Final answer:
To perform the indicated operation and simplify the result, we need to use the distributive property of multiplication over addition. The options can be solved as follows: a) -x^2 - 2x + 35, b) x^2 + 12x + 35, c) x^2 - 12x + 35, d) -x^2 + 2x + 35.
Step-by-step explanation:
To perform the indicated operation and simplify the result, we need to use the distributive property of multiplication over addition. Let's solve each option:
a) ({7 + {1}{x}}{5 - {1}{x}}) = 7(5 - {1}{x}) + {1}{x}(5 - {1}{x}) = 35 - 7x + 5x - x^2 = -x^2 - 2x + 35
b) ({7 + {1}{x}}{5 + {1}{x}}) = 7(5 + {1}{x}) + {1}{x}(5 + {1}{x}) = 35 + 7x + 5x + x^2 = x^2 + 12x + 35
c) ({7 - {1}{x}}{5 - {1}{x}}) = 7(5 - {1}{x}) - {1}{x}(5 - {1}{x}) = 35 - 7x - 5x + x^2 = x^2 - 12x + 35
d) ({7 - {1}{x}}{5 + {1}{x}}) = 7(5 + {1}{x}) - {1}{x}(5 + {1}{x}) = 35 + 7x - 5x - x^2 = -x^2 + 2x + 35