Question

Can you please tell me how to solve this????

Answer 1

Answer:

-2

Step-by-step explanation:

You have to simplify numerator and denominator through factoring. The you should get an expression involving only a + b which is given as -2

Take numerator
$a^2-b^2-12b-36$

Factor
$-b^2-12b-36$

$= - (b^2 + 12b + 36)\\\\= -(b+6)^2$

So numerator becomes
$a^2-\left(b+6\right)^2$

Apply difference of squares formula: $\displaystyle x^2-y^2=\left(x+y\right)\left(x-y\right)$

$\mathrm{with\;} x^2 == > a^2, and y^2 == > (b + 6)^2$

$\implies a^2-\left(b+6\right)^2 \\== \left(a+\left(b+6\right)\right)\left(a-b-6\right)$

Denominator
$a^2-6a-b^2-6b$

can be factored as follows

$a^2-b^2 = (a + b) (a -b)$
$-6a-6b = -6\left(a+b\right)$

So denominator becomes
$\left(a+b\right)\left(a-b\right)+-6\left(a+b\right)$

Factor out common term (a+b) to get
$\left(a+b\right)\left(a-b-6\right)$

So the original expression with Numerator and denominator

$\displaystyle =\frac{\left(a+b+6\right)\left(a-b-6\right)}{\left(a+b\right)\left(a-b-6\right)}$
Cancel the common factor (a-b-6) to get
$\displaystyle \frac{a+b+6}{a+b}$

Since a + b = -2, plug in this value of (a+b) in both numerator and denominator to get
$\displaystyle \dfrac{-2 + 6}{-2}$
$\displaystyle = - 2$

Answer: -2