Add the rational expressions. Write your answer in its fully factored form. (8)/(y^2)+17y+72+(4)/(y+8)

To add the rational expressions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators. <br /><br />The LCM of $y^2 + 17y + 72$ and $y + 8$ is $y^2 + 17y + 72$ because $y + 8$ is a factor of $y^2 + 17y + 72$.<br /><br />Now, we rewrite each fraction with the common denominator:<br /><br />$\frac{8}{y^2 + 17y + 72} + \frac{4(y + 8)}{y^2 + 17y + 72}$<br /><br />Next, we combine the numerators over the common denominator:<br /><br />$\frac{8 + 4(y + 8)}{y^2 + 17y + 72}$<br /><br />Simplify the numerator:<br /><br />$8 + 4y + 32 = 4y + 40$<br /><br />So, the expression becomes:<br /><br />$\frac{4y + 40}{y^2 + 17y + 72}$<br /><br />Factor out the common factor in the numerator:<br /><br />$4(y + 10)$<br /><br />Thus, the fully factored form of the expression is:<br /><br />$\frac{4(y + 10)}{y^2 + 17y + 72}$