The given expression is:
[tex]6+\lbrack(x+5)\div\frac{x^2+3x-10}{x-1}\rbrack[/tex]
Simplify the expression as follows:
Change the division sign to multiplication sign by swapping the denominator and numerator of the fraction on the right.
[tex]6+\lbrack(x+5)\times\frac{x-1}{x^2+3x-10}\rbrack[/tex]
Factorize the quadratic expression in the denominator:
[tex]\begin{gathered} 6+\lbrack(x+5)\times\frac{x-1}{x^2+3x-10}\rbrack_{} \\ \text{Rewrite 3x as 5x-2x, with the coefficients chosen such that their product is -10:} \\ =6+\lbrack(x+5)\times\frac{x-1}{x^2+5x-2x-10}\rbrack_{} \\ =6+\lbrack(x+5)\times\frac{x-1}{x(x+5)-2(x+5)}\rbrack_{}=6+\lbrack(x+5)\times\frac{x-1}{(x+5)(x-2)}\rbrack \end{gathered}[/tex]
Cancel out common factors:
[tex]\begin{gathered} 6+\lbrack\cancel{(x+5)}\times\frac{x-1}{\cancel{(x+5)}(x-2)}\rbrack \\ =6+\frac{x-1}{x-2} \end{gathered}[/tex]
Simplify the expression:
[tex]6+\frac{x-1}{x-2}=\frac{6(x-2)+(x-1)}{x-2}=\frac{6x-12+x-1}{x-2}=\frac{7x-13}{x-2}[/tex]
Hence, the expression has been simplified to the form (ax-b)/(cx-d), where a=7, b=13, c=1, and d=2.
The expression is simplified to:
[tex]\frac{7x-13}{x-2}[/tex]
