Please help! I've been having trouble solving linear equations and I need as much help as I can get! Please, go in to detail as much as you possibly can! Thank you!

This is basically a strightforward order of operations question. Let's go through it step-by-step.

3(g + 1)/8 = 5(g + 2)/7

3(g + 1) = 40(g + 2)/7 [multiplied both sides by 8]

21(g + 1) = 40(g + 2) [multiplied both sides by 7]

21g + 21 = 40(g + 2) [distributed 21 across the binomial (g + 1)]

21g + 21 = 40g + 80 [distributed 40 across the binomial (g + 2)]

-19g + 21 = 80 [subtracted 40g from both sides]

-19g = 59 [subtracted 21 from both sides]

g ≅ -3.1 [divided both sides by -19]