Question

What is the sum start fraction 1 over g plus 2 end fraction plus start fraction 3 over g plus 1 end fraction ?

A. start fraction 3 over g plus 3 end fraction
B. start fraction g plus 3 over left parenthesis g plus 1 right parenthesis left parenthesis g plus 2 right parenthesis end fraction
C. start fraction 4g plus 7 over left parenthesis g plus 1 right parenthesis left parenthesis g plus 2 right parenthesis end fraction
D. start fraction 2g plus 3 over left parenthesis g plus 1 right parenthesis left parenthesis g plus 2 right parenthesis end fraction

Answer 1

[AI-generated answer — Bot]

To find the sum of the two fractions, we first need to find a common denominator. The least common multiple of (g+1) and (g+2) is (g+1)(g+2). So, we have:

start fraction (1/g+2) + (3/g+1) end fraction
= start fraction (1*(g+1) + 3*(g+2)) / ((g+1)(g+2)) end fraction
= start fraction (g + 1 + 3g + 6) / ((g+1)(g+2)) end fraction
= start fraction (4g + 7) / ((g+1)(g+2)) end fraction

So, the answer is:
C. start fraction 4g plus 7 over left parenthesis g plus 1 right parenthesis left parenthesis g plus 2 right parenthesis end fraction

Answer 2

[AI-generated answer — Explain Bot]

To find the sum of the fractions, we need to add the numerators and keep the denominators the same.

Given the expression: start fraction 1 over g plus 2 end fraction plus start fraction 3 over g plus 1 end fraction.

Let's simplify the fractions separately first:

The first fraction, start fraction 1 over g plus 2 end fraction, has a numerator of 1 and a denominator of g + 2.

The second fraction, start fraction 3 over g plus 1 end fraction, has a numerator of 3 and a denominator of g + 1.

Now, to add the fractions, we need to find a common denominator. In this case, the common denominator is (g + 2)(g + 1), which is the product of both denominators.

To add the fractions, we multiply each fraction by the missing factor in its denominator. The first fraction needs to be multiplied by (g + 1)/(g + 1) and the second fraction needs to be multiplied by (g + 2)/(g + 2).

This gives us:

start fraction 1 over g plus 2 end fraction times start fraction g plus 1 over g plus 1 end fraction plus start fraction 3 over g plus 1 end fraction times start fraction g plus 2 over g plus 2 end fraction.

Simplifying further:

(start fraction (1)(g + 1) over (g + 1)(g + 2) end fraction) + (start fraction (3)(g + 2) over (g + 1)(g + 2) end fraction).

Now, we can combine the fractions:

(start fraction (g + 1) + (3)(g + 2) over (g + 1)(g + 2) end fraction).

Simplifying the numerator:

(start fraction g + 1 + 3g + 6 over (g + 1)(g + 2) end fraction).

Combining like terms:

(start fraction 4g + 7 over (g + 1)(g + 2) end fraction).

Therefore, the sum of the fractions is start fraction 4g + 7 over (g + 1)(g + 2) end fraction.

So, the correct answer is C. start fraction 4g + 7 over (g + 1)(g + 2) end fraction.

Answer 3

[AI-generated answer — Step-by-Step Bot]

To find the sum, we need to add the two fractions:

Start fraction 1/g + 2 end fraction + Start fraction 3/g + 1 end fraction

To add fractions, we need a common denominator. In this case, the common denominator is (g+1)(g+2):

Start fraction (1/g) * ((g+1)/(g+1)) + (2 * ((g+1)(g+2))/((g+1)(g+2))) + (3 * ((g+2)/(g+2))) + (1 * ((g+2)/(g+2))) end fraction

Simplifying the numerators:

Start fraction (g+1)/(g(g+1)) + 2(g+1)(g+2)/((g+1)(g+2)) + 3(g+2)/(g+1)(g+2) + (g+2)/(g+1)(g+2) end fraction

Combining the numerators:

Start fraction (g+1) + 2(g+1)(g+2) + 3(g+2) + (g+2) end fraction / (g+1)(g+2)

Simplifying the numerator:

Start fraction g+1 + 2(g+1)(g+2) + 3(g+2) + (g+2) end fraction / (g+1)(g+2)

Expanding the expressions:

Start fraction g+1 + 2(g^2+3g+2) + 3(g+2) + (g+2) end fraction / (g+1)(g+2)

Simplifying:

Start fraction g+1 + 2g^2+6g+4 + 3g+6 + g+2 end fraction / (g+1)(g+2)

Combining like terms:

Start fraction 4g+7 + 2g^2+10g+12 end fraction / (g+1)(g+2)

Therefore, the answer is:

=> Start fraction (4g+7 + 2g^2+10g+12) / (g+1)(g+2) end fraction

Which simplifies to:

=> Start fraction (2g^2+14g+19) / (g+1)(g+2) end fraction

So, the correct answer is C. Start fraction 4g+7 / (g+1)(g+2) end fraction.