Question

I didn’t understand this question, please help thank you very much

Answer 1

Hello there. To solve this question, we'll have to remember some properties about finding the common denominator of a sum of fractions.

Given the fractions:

`(5y^2+2)/(y^2+7y+10)+(3)/(y+2)`

We start by factoring the expression in the denominator of the first fraction.

Notice it is something of the form x² - Sx + P, where S = -7 and P = 10.

S and P are the sum and the product of the roots of the expression. In this case, we can easily find values that would give us a sum of -7 and a product of 10:

y = -2 and y = -5

Thus, we write it as:

y² + 7y + 10 = (y + 2)(y + 5)

Then we have:

`\frac{5y^2+2}{(y+2)(y+5)_{}}+(3)/(y+2)`

Now, to find the common denominator, we calculate the least common multiple of the expressions:

`\mathrm{lcm}((y+2)(y+5),(y+2))=(y+2)(y+5)`

We can calculate it as:

Therefore this is the common denominator of the fractions, in its factored form.