Simplify the expression completely:

4x^2+2x-42/4x /(x-3)

(x (8 x^2 - 20 x + -33))/(2 (x - 3))

Step-by-step explanation:

Simplify the following:

4 x^2 + 2 x - (42 x)/(4 (x - 3))

The gcd of 42 and 4 is 2, so 42/4 = (2×21)/(2×2) = 2/2×21/2 = 21/2:

4 x^2 + 2 x - x/(x - 3) 21/2

Put each term in 4 x^2 + 2 x - (21 x)/(2 (x - 3)) over the common denominator 2 (x - 3): 4 x^2 + 2 x - (21 x)/(2 (x - 3)) = (8 (x - 3) x^2)/(2 (x - 3)) + (4 (x - 3) x)/(2 (x - 3)) - (21 x)/(2 (x - 3)):

(8 x^2 (x - 3))/(2 (x - 3)) + (4 x (x - 3))/(2 (x - 3)) - (21 x)/(2 (x - 3))

(8 (x - 3) x^2)/(2 (x - 3)) + (4 (x - 3) x)/(2 (x - 3)) - (21 x)/(2 (x - 3)) = (8 (x - 3) x^2 + 4 (x - 3) x - 21 x)/(2 (x - 3)):

(8 x^2 (x - 3) + 4 x (x - 3) - 21 x)/(2 (x - 3))

Factor x out of 8 (x - 3) x^2 + 4 (x - 3) x - 21 x, resulting in x (8 (x - 3) x^(2 - 1) + 4 (x - 3) - 21):

(x (8 x^(2 - 1) (x - 3) + 4 (x - 3) - 21))/(2 (x - 3))

2 - 1 = 1:

(x (8 x (x - 3) + 4 (x - 3) - 21))/(2 (x - 3))

8 x (x - 3) = 8 x^2 - 24 x:

(x (8 x^2 - 24 x + 4 (x - 3) - 21))/(2 (x - 3))

4 (x - 3) = 4 x - 12:

(x (4 x - 12 + 8 x^2 - 24 x - 21))/(2 (x - 3))

Grouping like terms, 8 x^2 + 4 x - 24 x - 21 - 12 = 8 x^2 + (-24 x + 4 x) + (-12 - 21):

(x (8 x^2 + (-24 x + 4 x) + (-12 - 21)))/(2 (x - 3))

4 x - 24 x = -20 x:

(x (8 x^2 + -20 x + (-12 - 21)))/(2 (x - 3))

-12 - 21 = -33:

(x (8 x^2 - 20 x + -33))/(2 (x - 3))