answer：To find the coefficient of ( x ) in the expansion of ( left( x + frac{2}{x} right)^3 ), I will use the binomial theorem.First, I recall the binomial theorem, which states that ( (a + b)^n = sum_{k=0}^{n} binom{n}{k} a^{n-k} b^k ).Applying this theorem to ( left( x + frac{2}{x} right)^3 ), I identify ( a = x ), ( b = frac{2}{x} ), and ( n = 3 ).Next, I calculate each term in the expansion by substituting the values into the binomial formula. This gives me four terms corresponding to ( k = 0, 1, 2, 3 ).After expanding all the terms, I simplify each one and look for the term that contains ( x ). The term ( 6x ) is the only one with a single ( x ) factor.Finally, the coefficient of ( x ) in the expansion is 6.

answer：To determine the sum of the roots ( m ) and ( n ) of the quadratic equation ( x^2 + m x + n = 0 ), I can use Vieta's formulas. According to Vieta's formulas, the sum of the roots of a quadratic equation ( ax^2 + bx + c = 0 ) is given by ( -frac{b}{a} ).In this equation, ( a = 1 ), ( b = m ), and ( c = n ). Applying Vieta's formula, the sum of the roots is:[m + n = -frac{m}{1} = -m]This leads to the equation:[m + n = -m]Solving for ( n ), I get:[n = -2m]Next, using Vieta's formula for the product of the roots, which is ( frac{c}{a} ), I have:[m times n = n]Substituting ( n = -2m ) into this equation:[m times (-2m) = -2m^2 = n]Since ( n = -2m ), this implies:[-2m^2 = -2m]Dividing both sides by (-2):[m^2 = m]Subtracting ( m ) from both sides:[m^2 - m = 0]Factoring out ( m ):[m(m - 1) = 0]This gives two possible solutions for ( m ):[m = 0 quad text{or} quad m = 1]However, the problem states that ( m neq 0 ), so the valid solution is:[m = 1]Substituting ( m = 1 ) back into ( n = -2m ):[n = -2 times 1 = -2]Finally, the sum of the roots ( m + n ) is:[1 + (-2) = -1]

answer：First, I need to determine the cost of one pen, which is given as 2.Next, to find the cost of 10 pens, I multiply the cost of one pen by the number of pens: 10 pens × 2 per pen = 20.Therefore, the total cost of 10 pens is 20.

answer：First, I'll define the two positive integers as ( x ) and ( y ), where ( x > y ).Given that their difference is 12, I can write the equation:[ x - y = 12 ]I'll solve for ( x ) in terms of ( y ):[ x = y + 12 ]Next, since their product is 45, I'll set up the equation:[ x cdot y = 45 ]Substituting the expression for ( x ) from the first equation into the second equation gives:[ (y + 12) cdot y = 45 ][ y^2 + 12y = 45 ]To form a quadratic equation, I'll subtract 45 from both sides:[ y^2 + 12y - 45 = 0 ]Now, I'll solve this quadratic equation using the quadratic formula:[ y = frac{-b pm sqrt{b^2 - 4ac}}{2a} ]where ( a = 1 ), ( b = 12 ), and ( c = -45 ).Calculating the discriminant:[ sqrt{12^2 - 4 cdot 1 cdot (-45)} = sqrt{144 + 180} = sqrt{324} = 18 ]So, the solutions for ( y ) are:[ y = frac{-12 + 18}{2} = 3 ][ y = frac{-12 - 18}{2} = -15 ]Since we're looking for positive integers, ( y = 3 ).Substituting back to find ( x ):[ x = 3 + 12 = 15 ]Finally, the sum of the integers is:[ x + y = 15 + 3 = 18 ]

answer：To determine the value of ( t ) that makes the vectors ( overrightarrow{a} = (1, -1) ) and ( overrightarrow{b} = (2, t) ) perpendicular, I start by recalling that two vectors are perpendicular if their dot product is zero.First, I calculate the dot product of ( overrightarrow{a} ) and ( overrightarrow{b} ):[overrightarrow{a} cdot overrightarrow{b} = (1)(2) + (-1)(t) = 2 - t]Since the vectors are perpendicular, their dot product must equal zero:[2 - t = 0]Solving for ( t ), I find:[t = 2]

answer：First, I will simplify the expression ( a(a - 2b) + (a + b)^2 - (a + b)(a - b) ).Expanding ( a(a - 2b) ) gives ( a^2 - 2ab ).Expanding ( (a + b)^2 ) results in ( a^2 + 2ab + b^2 ).Expanding ( (a + b)(a - b) ) yields ( a^2 - b^2 ).Substituting these expansions back into the original expression:[a^2 - 2ab + a^2 + 2ab + b^2 - (a^2 - b^2)]Combining like terms:[2a^2 + b^2 - a^2 + b^2 = a^2 + 2b^2]Now, substituting ( a = 1 ) and ( b = -2 ) into the simplified expression:[1^2 + 2(-2)^2 = 1 + 8 = 9]