Question

What is the simplified form of the quantity of x plus 5, all over the quantity of 3x plus 4 + the quantity of x plus 4, all over the quantity of x plus 3?

Answer 1

`(x + 5)/(3x + 4) + (x + 4)/(x + 3) = ((x + 5)(x + 3))/((3x + 4)(x + 3)) + ((x + 4)(3x + 4))/((3x + 4)(x + 3)) = (x(x + 3) + 5(x + 3))/((3x(x + 3) + 4(x + 3)) + (x(3x + 4) + 4(3x + 4))/(3x(x + 3) + 4(x + 3)) = (x(x) + x(3) + 5(x) + 5(3))/(3x(x) + 3x(3) + 4(x) + 4(3)) + (x(3x) + x(4) + 4(3x) + 4(4))/(3x(x) + 3x(3) + 4(x) + 4(3)) = (x^(2) + 3x + 5x + 15)/(3x^(2) + 9x + 4x + 12) + (3x^(2) + 4x + 12x + 16)/(3x^(2) + 9x + 4x + 12) = (x^(2) + 8x + 15)/(3x^(2) + 13x + 12) + (3x^(2) + 16x + 16)/(3x^(2) + 13x + 12) = ((x^(2) + 8x + 15) + (3x^(2) + 16x + 16))/(3x^(2) + 13x + 12) = ((x^(2) + 3x^(2)) + (8x + 16x) + (15 + 16))/(3x^(2) + 13x + 12) = (4x^(2) + 24x + 31)/(3x^(2) + 13x + 12)`

Answer 2

Answer:
`(4x^2+24x+31)/(3x^2+13x+12)`

Explanation:

Since the given expression ,

`(x+5)/(3x+4) + (x+4)/(x+3)`

=
`((x+5)(x+3)+(x+4)(3x+4))/((3x+4)(x+3))` ( by taking the LCM for solving the fraction)

=
`(x^2+5x+3x+15+3x^2+12x+4x+16)/(3x^2+9x+4x+12)` ( By multiplying)

=
`(x^2+8x+15+3x^2+16x+16)/(3x^2+13x+12)`

=
`(4x^2+24x+31)/(3x^2+13x+12)` ( by adding the like terms)