Final answer:
To simplify the expression, factor the denominators and find a common denominator which is (t+3)(t+5)(t-2). Rewrite the fractions with this common denominator, simplify the resulting numerator, and then place it over the common denominator.
Step-by-step explanation:
The student has asked for help with the following operation:
(3)/(t²+t-6)+(1)/(t²+3t-10)
To simplify the given expression, we should first factor the denominators to find a common denominator. The denominators are quadratic expressions, so we will factor each one.
The first denominator t²+t-6 factors into (t+3)(t-2), and the second denominator t²+3t-10 factors into (t+5)(t-2). The common denominator is (t+3)(t+5)(t-2).
Now, we rewrite each fraction with the common denominator:
(3/((t+3)(t-2)))*((t+5)/(t+5)) + (1/((t+5)(t-2)))*((t+3)/(t+3))
After multiplying through, we obtain:
(3(t+5))/((t+3)(t+5)(t-2)) + (1(t+3))/((t+3)(t+5)(t-2))
The combined numerator is 3(t+5) + t+3. Simplify the numerator and then place it over the common denominator to complete the simplification.
Finally, the simplified form is:
(3t+15+t+3)/((t+3)(t+5)(t-2)) = (4t+18)/((t+3)(t+5)(t-2))
If possible, we should check for further simplification, but since 4t+18 does not have (t+3), (t+5), or (t-2) as a factor, there's no further simplification needed.