Solve 7/a^2+6a+9-5/a^2-a-12=

$$\frac { 7 } { a ^ { 2 } + 6 a + 9 } - \frac { 5 } { a ^ { 2 } - a - 12 } =$$

Evaluate

$$\frac{2a-43}{\left(a-4\right)\left(a+3\right)^{2}}$$

Short Solution Steps

Factor a^{2}+6a+9. Factor a^{2}-a-12.

$$\frac{7}{\left(a+3\right)^{2}}-\frac{5}{\left(a-4\right)\left(a+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+3\right)^{2} and \left(a-4\right)\left(a+3\right) is \left(a-4\right)\left(a+3\right)^{2}. Multiply \frac{7}{\left(a+3\right)^{2}} times \frac{a-4}{a-4}. Multiply \frac{5}{\left(a-4\right)\left(a+3\right)} times \frac{a+3}{a+3}.

$$\frac{7\left(a-4\right)}{\left(a-4\right)\left(a+3\right)^{2}}-\frac{5\left(a+3\right)}{\left(a-4\right)\left(a+3\right)^{2}}$$

Since \frac{7\left(a-4\right)}{\left(a-4\right)\left(a+3\right)^{2}} and \frac{5\left(a+3\right)}{\left(a-4\right)\left(a+3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7\left(a-4\right)-5\left(a+3\right)}{\left(a-4\right)\left(a+3\right)^{2}}$$

Do the multiplications in 7\left(a-4\right)-5\left(a+3\right).

$$\frac{7a-28-5a-15}{\left(a-4\right)\left(a+3\right)^{2}}$$

Combine like terms in 7a-28-5a-15.

$$\frac{2a-43}{\left(a-4\right)\left(a+3\right)^{2}}$$

Expand \left(a-4\right)\left(a+3\right)^{2}.

$$\frac{2a-43}{a^{3}+2a^{2}-15a-36}$$

Differentiate w.r.t. a

$$-\frac{\left(8a-137\right)^{2}-14945}{16\left(a-4\right)^{2}\left(a+3\right)^{3}}$$