express the area of the entire rectangle. your answer should be a polynomial in standard form. area =

Explanation:

Step1: Recall area - formula for rectangle

The area of a rectangle is length times width. Here, the length of the rectangle is (b^{2}+7b + 4) and the width is (b^{2}+b).

Step2: Use the distributive property (FOIL - like for polynomials)

((b^{2}+b)(b^{2}+7b + 4)=b^{2}(b^{2}+7b + 4)+b(b^{2}+7b + 4))

Step3: Distribute (b^{2}) and (b)

[ \begin{align*} b^{2}(b^{2}+7b + 4)&=b^{2}\times b^{2}+b^{2}\times7b + b^{2}\times4=b^{4}+7b^{3}+4b^{2}\ b(b^{2}+7b + 4)&=b\times b^{2}+b\times7b + b\times4=b^{3}+7b^{2}+4b \end{align*} ]

Step4: Combine like - terms

[ \begin{align*} (b^{4}+7b^{3}+4b^{2})+(b^{3}+7b^{2}+4b)&=b^{4}+(7b^{3}+b^{3})+(4b^{2}+7b^{2})+4b\ &=b^{4}+8b^{3}+11b^{2}+4b \end{align*} ]

Answer:

(b^{4}+8b^{3}+11b^{2}+4b)