Perform the indicated operations, then simplify \( \frac{5 y}{y+6}+\frac{7 y^{2}}{y^{2}-36}-\frac{4}{y-6} \) \( \frac{12 y^{2}-34 y+24}{(y-6)(y+6)} \) \( \frac{12 y^{2}-34 y-24}{(y+6)(y-6)} \) \( \frac{7 y^{2}+5 y-4}{(y+6)(y-6)} \) \( \frac{7 y^{2}+5 y-4}{y^{2}+2 y-36} \)

To simplify the expression \[ \frac{5y}{y+6} + \frac{7y^2}{y^2-36} - \frac{4}{y-6}, \] we first need to find a common denominator. The denominators are \(y + 6\), \(y^2 - 36\), and \(y - 6\). Notice that \(y^2 - 36\) can be factored as \((y - 6)(y + 6)\). Therefore, the common denominator for all three fractions is \((y - 6)(y + 6)\). Now, we rewrite each term with the common denominator: 1. For \(\frac{5y}{y + 6}\): \[ \frac{5y}{y + 6} = \frac{5y(y - 6)}{(y + 6)(y - 6)} = \frac{5y^2 - 30y}{(y - 6)(y + 6)}. \] 2. For \(\frac{7y^2}{y^2 - 36}\): \[ \frac{7y^2}{y^2 - 36} = \frac{7y^2}{(y - 6)(y + 6)} = \frac{7y^2}{(y - 6)(y + 6)}. \] 3. For \(-\frac{4}{y - 6}\): \[ -\frac{4}{y - 6} = -\frac{4(y + 6)}{(y - 6)(y + 6)} = \frac{-4y - 24}{(y - 6)(y + 6)}. \] Now we can combine all three fractions: \[ \frac{5y^2 - 30y + 7y^2 - 4y - 24}{(y - 6)(y + 6)}. \] Combine the numerators: \[ 5y^2 + 7y^2 - 30y - 4y - 24 = 12y^2 - 34y - 24. \] Thus, we have: \[ \frac{12y^2 - 34y - 24}{(y - 6)(y + 6)}. \] Next, we can factor the numerator \(12y^2 - 34y - 24\). We can factor out a common factor of 2: \[ 12y^2 - 34y - 24 = 2(6y^2 - 17y - 12). \] Now we need to factor \(6y^2 - 17y - 12\). We look for two numbers that multiply to \(6 \times -12 = -72\) and add to \(-17\). The numbers \(-24\) and \(3\) work: \[ 6y^2 - 24y + 3y - 12 = 6y(y - 4) + 3(y - 4) = (6y + 3)(y - 4). \] Thus, we have: \[ 12y^2 - 34y - 24 = 2(6y + 3)(y - 4). \] Putting it all together, we have: \[ \frac{2(6y + 3)(y - 4)}{(y - 6)(y + 6)}. \] This is the simplified form of the original expression. So the final answer is: \[ \frac{2(6y + 3)(y - 4)}{(y - 6)(y + 6)}. \]