Which expression represents the sum of the following?
5 / 2t + 1 + -1 / 4t^2 + 4t + 1
a)10t + 4 / (2t + 1)^2
b) 4 / (2t + 1)^3
c)10t + 1 / (2t + 1)^2
d) 4 / 4t^2 + 6t + 2
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To find the sum of the given expressions, we need to combine the like terms.
The given expression is:
5 / 2t + 1 + -1 / 4t^2 + 4t + 1
First, let's combine the like terms in the numerator of the first fraction: 5 / 2t + 1
Since there are no other terms in the numerator with 't', this term remains as it is.
Now, let's combine the like terms in the numerator of the second fraction: -1 / 4t^2 + 4t + 1
We can't combine these terms further, so they remain as they are.
Thus, the sum of the expressions becomes:
5 / 2t + 1 + -1 / 4t^2 + 4t + 1
Now, we can rewrite the sum of the expressions in a single fraction by finding the common denominator.
The common denominator is (2t + 1)^2 since it is present in both fractions.
Multiplying the first fraction by (2t + 1) / (2t + 1) and the second fraction by (2t + 1) / (2t + 1), we get:
(5 * (2t + 1) + -1) / (2t + 1)^2 + (4t^2 + 4t + 1) / (2t + 1)^2
Now, let's simplify the numerator:
(10t + 5 - 1) / (2t + 1)^2 + (4t^2 + 4t + 1) / (2t + 1)^2
(10t + 4) / (2t + 1)^2 + (4t^2 + 4t + 1) / (2t + 1)^2
Combining the fractions with the common denominator, we get:
(10t + 4 + 4t^2 + 4t + 1) / (2t + 1)^2
(4t^2 + 14t + 5) / (2t + 1)^2
Comparing the given options:
a) 10t + 4 / (2t + 1)^2
b) 4 / (2t + 1)^3
c) 10t + 1 / (2t + 1)^2
d) 4 / 4t^2 + 6t + 2
The simplified sum of the expressions matches the option (c) 10t + 1 / (2t + 1)^2.
Therefore, the answer is option (c) 10t + 1 / (2t + 1)^2.