Question

1. $\frac{9x^{2}+6x+1}{x+5} div \frac{3x+1}{x^{2}+5x}$

Explanation:

Step1: Rewrite division as multiplication

$\frac{9x^2+6x+1}{x+5} \times \frac{x^2+5x}{3x+1}$

Step2: Factor all expressions

$9x^2+6x+1=(3x+1)^2$, $x^2+5x=x(x+5)$
Substitute factored forms:
$\frac{(3x+1)^2}{x+5} \times \frac{x(x+5)}{3x+1}$

Step3: Cancel common factors

Cancel $(3x+1)$ and $(x+5)$:
$(3x+1) \times x$

Step4: Simplify the product

$x(3x+1)=3x^2+x$
Note: $x
eq -5, x
eq -\frac{1}{3}, x
eq 0$ (values that make original denominators zero)

Answer:

$3x^2 + x$ (where $x
eq -5, -\frac{1}{3}, 0$)