Question

a rectangle has a height of $x + 4$ and a width of $x^{2}+3x + 2$. express the area of the entire rectangle. your answer should be a polynomial in standard form.

Explanation:

Step1: Recall area formula

The area $A$ of a rectangle is $A = \text{height}\times\text{width}$. Here, height $h=x + 4$ and width $w=x^{2}+3x + 2$. So $A=(x + 4)(x^{2}+3x + 2)$.

Step2: Use distributive property

$(x + 4)(x^{2}+3x + 2)=x(x^{2}+3x + 2)+4(x^{2}+3x + 2)$.

Step3: Distribute $x$ and $4$

$x(x^{2}+3x + 2)=x^{3}+3x^{2}+2x$ and $4(x^{2}+3x + 2)=4x^{2}+12x + 8$.

Step4: Combine like - terms

$A=(x^{3}+3x^{2}+2x)+(4x^{2}+12x + 8)=x^{3}+(3x^{2}+4x^{2})+(2x + 12x)+8=x^{3}+7x^{2}+14x + 8$.

Answer:

$x^{3}+7x^{2}+14x + 8$