Question

1. $(x^{3}+4x^{2}+4x)\div(x + 2)$

Explanation:

Step1: Factor the numerator

First, factor out the common term $x$ from the polynomial, then factor the quadratic.
$x^3 + 4x^2 + 4x = x(x^2 + 4x + 4) = x(x+2)^2$

Step2: Rewrite the division

Replace the numerator with its factored form and rewrite division as multiplication by reciprocal.
$\frac{x(x+2)^2}{x+2}$

Step3: Simplify the expression

Cancel the common factor $(x+2)$ (where $x
eq -2$).
$x(x+2) = x^2 + 2x$

Answer:

$x^2 + 2x$ (where $x
eq -2$)