1) Let's factorize in $ℝ$: $11x^3$$+125x^2$$+45x$$+11$

⇥1.1) Try the obvious solution first: $x=-11$

⇥Let's evaluate the value of: $11·{\left(-11\right)}^3$$+125·{\left(-11\right)}^2$$+45·\left(-11\right)$$+11$

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product: ${\left(-11\right)}^3$ = $\left(-11\right)·\left(-11\right)·\left(-11\right)$ = $-1331$, ${\left(-11\right)}^2$ = $\left(-11\right)·\left(-11\right)$ = $121$■Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative: $45·\left(-11\right)$ = $-45·11$ = $-45·(10+1)$ = $-(45·10+45)$ = $-(450+45)$ = $-495$

⇥ $11·\left(-1331\right)$$+125·121$$-495$$+11$

⇥ⓘ Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative: $11·\left(-1331\right)$ = $-11·1331$ = $-(10+1)·1331$ = $-(10·1331+1331)$ = $-(\mathrm{13,310}+1331)$ = $\mathrm{-14,641}$■Multiplications are done before additions and subtractions: $125·121$ = $125·(100+20+1)$ = $125·100+125·20+125$ = $\mathrm{12,500}+2500+125$ = $\mathrm{15,125}$■Start from $-495$ and move forward $11$ on the number line, getting closer to zero: $-495+11$ = $-484$

⇥ $-\mathrm{14,641}+\mathrm{15,125}-484$

⇥ⓘ Start from $\mathrm{-14,641}$ and move forward $\mathrm{15,125}$ on the number line, going past zero: $-\mathrm{14,641}+\mathrm{15,125}$ = $484$

⇥ $484-484$

⇥ⓘ Start from $484$ and move back $484$ on the number line, down to zero: $484-484$ = $0$

⇥ $0$

⇥Root found: $x=-11$

⇥1.2) Compute the quotient and remainder of $(11x^3$$+125x^2$$+45x$$+11)$ ÷ $(x+11)$

⇥

Expression in progress	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$11x^3$$+125x^2$$+45x$$+11$	$\frac{11x^3}{x}$	$11x^2$	$11x^2·(x+11)=11x^3+121x^2$	$(11x^3$$+125x^2$$+45x$$+11)-(11x^3+121x^2)=4x^2+45x+11$
$4x^2+45x+11$	$\frac{4x^2}{x}$	$4x$	$4x·(x+11)=4x^2+44x$	$(4x^2+45x+11)-(4x^2+44x)=x+11$
$x+11$	$\frac{x}{x}$	$1$	$x+11=x+11$	$(x+11)-(x+11)=0$

⇥Therefore, the result of the euclidean division is: $11x^2+4x+1$

ⅰ $11x^3$$+125x^2$$+45x$$+11$ = $(x+11)·(11x^2+4x+1)$

▶Factored expression:   $(x+11)·(11x^2+4x+1)$