answer:Let's analyze the function ( f(x) = x^4 + x^2 + 5x + 3 ). We can express ( f(x) ) as the sum of an even function ( g(x) = x^4 + x^2 + 3 ) and an odd function ( h(x) = 5x ). Since ( g(x) ) is even, ( g(x) = g(-x) ), and since ( h(x) ) is odd, ( h(-x) = -h(x) ). Therefore, we can simplify ( f(x) - f(-x) ) as follows: [ f(x) - f(-x) = (g(x) + h(x)) - (g(-x) + h(-x)) = g(x) - g(-x) + h(x) - h(-x) = 0 + 2h(x) = 2 times 5x ] Substituting ( x = 5 ) into the simplified expression: [ f(5) - f(-5) = 2 times 5 times 5 = 50 ] Thus, ( f(5) - f(-5) = boxed{50} ).

answer:# Solution: Part (1): Given the "beautiful two-variable linear equation" 5x-y=1, we substitute y=2x to find the "beautiful value": begin{align*} 5x - y &= 1 5x - 2x &= 1 3x &= 1 x &= frac{1}{3} end{align*} Thus, the "beautiful value" is boxed{frac{1}{3}}. Part (2): Given the "beautiful two-variable linear equation" frac{1}{3}x+y=m, we substitute y=2x and x=-3 to find m: begin{align*} frac{1}{3}x + y &= m frac{1}{3}(-3) + 2(-3) &= m -1 - 6 &= m m &= -7 end{align*} Therefore, the value of m is boxed{-7}. Part (3): For the equation frac{5}{2}x+y=n, substituting y=2x gives: begin{align*} frac{5}{2}x + 2x &= n frac{9}{2}x &= n x &= frac{2n}{9} end{align*} The "beautiful value" here is frac{2n}{9}. For the equation 4x-y=n-2, substituting y=2x gives: begin{align*} 4x - 2x &= n - 2 2x &= n - 2 x &= frac{n - 2}{2} end{align*} The "beautiful value" here is frac{n - 2}{2}. Equating the "beautiful values" to find n: begin{align*} frac{2n}{9} &= frac{n - 2}{2} 4n &= 9(n - 2) 4n &= 9n - 18 5n &= 18 n &= frac{18}{5} end{align*} Substituting n = frac{18}{5} into either "beautiful value" equation, we find x: begin{align*} x &= frac{2n}{9} = frac{2(frac{18}{5})}{9} = frac{4}{5} end{align*} Therefore, the value of n is boxed{frac{18}{5}} and the "beautiful value" at that time is boxed{frac{4}{5}}.

answer:Let's denote the number of boys as B and the number of girls as G. According to the given ratio, we have: B : G = 3 : 5 This means that for every 3 boys, there are 5 girls. To find the actual numbers, we need to consider the total number of students in the school, which is 480. The total parts of the ratio are 3 (for boys) + 5 (for girls) = 8 parts. Now, we can find the value of each part by dividing the total number of students by the total number of parts: Each part = Total students / Total parts Each part = 480 / 8 Each part = 60 Now that we know the value of each part, we can find the number of girls by multiplying the number of parts for girls by the value of each part: Number of girls (G) = Number of parts for girls × Value of each part G = 5 × 60 G = 300 So, there are boxed{300} girls in the school.

answer:Given the function f(x)=sin (2x+ frac {pi}{6})+cos (2x- frac {pi}{3}), we can rewrite it as f(x)=sin 2xcos frac {pi}{6}+cos 2xsin frac {pi}{6}+cos 2xcos frac {pi}{3}+sin 2xsin frac {pi}{3} = sqrt {3}sin 2x+cos 2x=2sin (2x+ frac {pi}{6}). From - frac {pi}{2}+2kpileqslant 2x+ frac {pi}{6}leqslant frac {pi}{2}+2kpi (kinmathbb{Z}), we get - frac {2pi}{3}+2kpileqslant 2xleqslant frac {pi}{3}+2kpi, (kinmathbb{Z}), Thus, - frac {pi}{3}+kpileqslant xleqslant frac {pi}{6}+kpi, (kinmathbb{Z}), Therefore, the interval of monotonic increase for f(x) is left[- frac {pi}{3}+kpi, frac {pi}{6}+kpiright], (kinmathbb{Z}), Hence, the correct option is boxed{text{D}}. This problem involves simplifying the expression using the sine of sum and cosine of difference formulas, and determining the answer through the overall concept and the increasing interval of the sine function. It examines the increasing interval of the sine function, as well as the application of the sine of sum and cosine of difference formulas, and the overall concept.