What is the product? x^2-16/2x+8 multiplied by x^3-2x^2+x/x^2+3x-4

(x (x - 4) (x - 1))/(2 (x + 4))

Step-by-step explanation:

Simplify the following:

((x^2 - 16) (x^3 - 2 x^2 + x))/((2 x + 8) (x^2 + 3 x - 4))

The factors of -4 that sum to 3 are 4 and -1. So, x^2 + 3 x - 4 = (x + 4) (x - 1):

((x^2 - 16) (x^3 - 2 x^2 + x))/((x + 4) (x - 1) (2 x + 8))

Factor 2 out of 2 x + 8:

((x^2 - 16) (x^3 - 2 x^2 + x))/(2 (x + 4) (x + 4) (x - 1))

x^2 - 16 = x^2 - 4^2:

((x^2 - 4^2) (x^3 - 2 x^2 + x))/(2 (x + 4) (x + 4) (x - 1))

Factor the difference of two squares. x^2 - 4^2 = (x - 4) (x + 4):

((x - 4) (x + 4) (x^3 - 2 x^2 + x))/(2 (x + 4) (x + 4) (x - 1))

Factor x out of x^3 - 2 x^2 + x:

(x (x^2 - 2 x + 1) (x - 4) (x + 4))/(2 (x + 4) (x + 4) (x - 1))

The factors of 1 that sum to -2 are -1 and -1. So, x^2 - 2 x + 1 = (x - 1) (x - 1):

(x (x - 1) (x - 1) (x - 4) (x + 4))/(2 (x + 4) (x + 4) (x - 1))

(x - 1) (x - 1) = (x - 1)^2:

(x (x - 1)^2 (x - 4) (x + 4))/(2 (x + 4) (x + 4) (x - 1))

((x - 4) (x + 4) x (x - 1)^2)/(2 (x + 4) (x + 4) (x - 1)) = (x + 4)/(x + 4)×((x - 4) x (x - 1)^2)/(2 (x + 4) (x - 1)) = ((x - 4) x (x - 1)^2)/(2 (x + 4) (x - 1)):

(x (x - 4) (x - 1)^2)/(2 (x + 4) (x - 1))

Cancel terms. ((x - 4) x (x - 1)^2)/(2 (x + 4) (x - 1)) = ((x - 4) x (x - 1)^(2 - 1))/(2 (x + 4)):

(x (x - 4) (x - 1)^(2 - 1))/(2 (x + 4))

2 - 1 = 1:

(x (x - 4) (x - 1))/(2 (x + 4))

Step-by-step explanation:

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b. an equilateral triangle inscribed in a circle

step-by-step explanation:

we are given the figure of two circles having a triangle inside.

now, we know that,

"when a figure is lying inside an another figure, then the outside figure is known as circumscribed figure and the figure lying inside is referred to as the inscribed figure".

as we see that, the circles are containing the triangle completely.

thus, the circles are the circumscribed figures and the triangle is the inscribed figure.

hence, we get, 'an equilateral triangle is inscribed in a circle'.