Question

question
fully simplify the expression below and write your answer as a single fraction.
$\frac{x^{4}+14x^{3}+40x^{2}}{x^{2}+9x+20} cdot \frac{3x+15}{x^{2}+11x+10}$

Explanation:

Step1: Factor all polynomials

1. Numerator 1: $x^4 + 14x^3 + 40x^2 = x^2(x^2 + 14x + 40) = x^2(x + 4)(x + 10)$
2. Denominator 1: $x^2 + 9x + 20 = (x + 4)(x + 5)$
3. Numerator 2: $3x + 15 = 3(x + 5)$
4. Denominator 2: $x^2 + 11x + 10 = (x + 1)(x + 10)$

Step2: Substitute factored forms

$$\frac{x^2(x + 4)(x + 10)}{(x + 4)(x + 5)} \cdot \frac{3(x + 5)}{(x + 1)(x + 10)}$$

Step3: Cancel common factors

Cancel $(x+4)$, $(x+5)$, $(x+10)$ from numerator and denominator:
$$\frac{x^2 \cdot 3}{x + 1}$$

Step4: Rewrite simplified expression

$\frac{3x^2(x + 4)}{x + 1}$ (corrected to $\frac{3x^2}{x + 1}$ with note that $(x+4)$ was canceled in step 3, final simplified form is $\frac{3x^2}{x + 1}$)

Correction note: In step 3, $(x+4)$ is canceled, so the final simplified form is $\frac{3x^2}{x + 1}$

Final Answer (corrected):

$\frac{3x^2}{x + 1}$

(corrected):

Step1: Factor all polynomials

1. $x^4 + 14x^3 + 40x^2 = x^2(x + 4)(x + 10)$
2. $x^2 + 9x + 20 = (x + 4)(x + 5)$
3. $3x + 15 = 3(x + 5)$
4. $x^2 + 11x + 10 = (x + 1)(x + 10)$

Step2: Substitute factored forms

$$\frac{x^2(x + 4)(x + 10)}{(x + 4)(x + 5)} \cdot \frac{3(x + 5)}{(x + 1)(x + 10)}$$

Step3: Cancel common terms

Cancel $(x+4)$, $(x+5)$, $(x+10)$:
$$\frac{x^2 \cdot 3}{x + 1}$$

Step4: Write final simplified fraction

$\frac{3x^2}{x + 1}$

Answer:

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