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Simplifying Fractions

Illustrations:

Simplify the following fractions:

1)

$$\frac{a^2+bc+ab+ac}{ax^2+by^2+bx^2+ay^2}=\frac{a^2+ab+ac+bc}{ax^2+ay^2+bx^2+by^2}$$

$$ =\frac{a(a+b)+c(a+b)}{a(x^2+y^2)+b(x^2+y^2) }$$

$$=\frac{(a+b)(a+c)}{(x^2+y^2)(a+ b)}$$

$$=\frac{(a+c)}{(x^2+y^2)}$$

Therefore $$\frac{a^2+bc+ab+ac}{ax^2+by^2+bx^2+ay^2} = \frac{(a+c)}{(x^2+y^2)}$$

2)

$$\frac{4x^2+12xy+9y^2 }{4x^2-9y^2 }$$

$$=\frac{(2x)^2+2(2x)(3y)+(3y)^2 }{(2x)^2-(3y)^2 }$$

$$=\frac{(2x+3y)^2 }{(2x+3y)(2x-3y) }$$

$$=\frac{(2x+3y)(2x+3y) }{(2x+3y)(2x-3y) }$$

$$=\frac{(2x+3y) }{(2x-3y) }$$