What is the least common denominator of the rational expressions below? 1/x^2-5x+3/x^2+2x-35

Here the denominators are quadratics, so we need to factorise them first

1 / (x²-5x) + 3 / (x²+2x-35)

factorising x²-5x,

here GCF is x, so taking x out we have

x (x-5)

factorising x²+2x-35 using AC method,

we find factors of - 35 that add up to give 2

(x-5) (x+7)

Since we have x-5 common in both denominators, so lcd of the rational expression will be product of x (x-5) (x+7)

lcd is x (x-5) (x+7)