36. $$ \frac{3 a^{2}-5 a b-12 b^{2}}{3 a b+4 b^{2}} \div\left(3 b^{2}-a b\right) $$

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UpStudy AI · Accepted Answer

To simplify the given expression, we will first simplify the numerator and denominator separately, and then perform the division.

Given expression:
$$ \frac{3 a^{2}-5 a b-12 b^{2}}{3 a b+4 b^{2}} \div\left(3 b^{2}-a b\right) $$

Simplify the numerator:
$$ 3 a^{2}-5 a b-12 b^{2} $$

Simplify the denominator:
$$ 3 a b+4 b^{2} $$

Now, we will simplify the expression by dividing the numerator by the denominator and then dividing the result by $$3 b^{2}-a b$$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\frac{\left(3a^{2}-5ab-12b^{2}\right)}{\left(3ab+4b^{2}\right)}}{\left(3b^{2}-ab\right)}$$
- step1: Remove the parentheses:
  $$\frac{\frac{3a^{2}-5ab-12b^{2}}{3ab+4b^{2}}}{3b^{2}-ab}$$
- step2: Divide the terms:
  $$\frac{\frac{a-3b}{b}}{3b^{2}-ab}$$
- step3: Multiply by the reciprocal:
  $$\frac{a-3b}{b}\times \frac{1}{3b^{2}-ab}$$
- step4: Rewrite the expression:
  $$\frac{-3b+a}{b}\times \frac{1}{-b\left(-3b+a\right)}$$
- step5: Reduce the fraction:
  $$\frac{1}{b}\times \frac{1}{-b}$$
- step6: Multiply the terms:
  $$-\frac{1}{b\times b}$$
- step7: Multiply the terms:
  $$-\frac{1}{b^{2}}$$
The simplified expression is $$ -\frac{1}{b^{2}} $$.
The simplified expression is $$ -\frac{1}{b^{2}} $$.

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