Can someone please explain how to solve this with cross multiplication? (x+2)/(x+3) = (x-7)/(x+1)?

(x+2)/(x+3)=(x-7)/(x+1). Multiply through by (x+3)(x+1) getting (x+2)(x+1)=(x+3)(x-7);

Expanding both sides gives x^2+3x+2 = x^2-4x-21. Now, going term by term; subtract RS from LS getting (x^2-x^2)+(3x-[-4x])+( 2-[-21])=0, ie., 7x+23=0. Then;

x = - (23/7).

(x+2)/(x+3) = (x-7)/(x+1)

Eliminate the fractions by multiplying by (x + 3)

x + 2 = (x + 3)(x - 7)/(x + 1)

and (x + 1)

(x +1) (x + 2) = (x + 3)(x - 7)

multiply out the brackets

x² + 3x + 2 = x² - 4x - 21

bring all the x values to the left of equal sign (remember,they change signs)

7x = -23

x = - 23/7

(x + 2)/(x + 3) = (x - 7)/(x + 1)

(x + 2)(x + 1) = (x + 3)(x - 7)

x^2 + 3x + 2 = x^2 - 4x - 21

7x = -23

x = -23/7

Personally, I prefer to phrase it "maintain equality"

Multiply both sides of equal sign by same amount (x+1) and (x+3)

Expand and collect terms, solve for 'x'

x =  -23/7

If: (x+2)/(x+3) = (x-7)/(x+1)

Then: (x+2)(x+1) = (x+3)(x-7)

Multiply out the brackets: x^2 +3x +2 = x^2 -4x -21

Collecting like terms: 7x = -23

Divide both sides by 7: x = -23/7

The cross multiplication part was simply 

(x + 2)(x + 1) = (x - 7)(x + 3), observe that terms in x^2 cancel 

3x + 2 = - 4x - 21. 

x = -23/7 

(x + 2)/(x + 3) = (x - 7)/(x + 1) <=== Note: x + 1 ≠ 0, x ≠ -1 ∪ x + 3 ≠ 0, x ≠ -3.

(x + 2) * (x + 1) = (x - 7) * (x + 3).

x^2 + 3x + 2 = x^2 - 4x - 21.

x^2 - x^2 + 3x + 4x = -2 - 21.

7x = -23.

x = -23/7... Or, alternatively: x = -3 2/7 (feasible solution).