The result after simplifying the expression is [tex]\frac{12}{x^2-2x-8}[/tex]
Given the difference of function expressed as:
[tex]\frac{x^2-3x+2}{x^2-5x+4} -\frac{x^2+10x+24}{x^2+8x+12} \\[/tex]
Factorize the quadratic equations:
[tex]=\frac{x^2-2x-x+2}{x^2-4x-x+4} -\frac{x^2+4x+6x+24}{x^2+6x+2x+12} \\=\frac{x(x-2)-1(x-2)}{x(x-4)-1(x-4)} -\frac{x(x+4)+6(x+4)}{x(x+6)+2(x+6)}\\=\frac{(x-1)(x-2)}{(x-1(x-4)} -\frac{(x+6)(x+4)}{(x+2)(x+6)}[/tex]
Cancel out the common terms:
[tex]\frac{(x-2)}{(x-4)} -\frac{(x+4)}{(x+2)}[/tex]
Find the LCM of the resulting expression
[tex]=\frac{(x-2)}{(x-4)} -\frac{(x+4)}{(x+2)}\\=\frac{(x+2)(x-2)-[(x-4)(x+4)]}{(x-4)(x+2)} \\=\frac{x^2-4-(x^2-16)}{(x-4)(x+2)} \\=\frac{x^2-4-x^2+16}{(x^2+2x-4x-8)}\\=\frac{12}{x^2-2x-8}[/tex]
Hence the result after simplifying the expression is [tex]\frac{12}{x^2-2x-8}[/tex]
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