Question

a rectangle has a height of $k^{2}+3$ and a width of $k^{2}+7$. express the area of the entire rectangle. your answer should be a polynomial in standard form. area =

Explanation:

Step1: Recall area formula

The area $A$ of a rectangle is given by $A = \text{height}\times\text{width}$. Here, height $h=k^{2}+3$ and width $w = k^{2}+7$. So $A=(k^{2}+3)(k^{2}+7)$.

Step2: Use FOIL method

$(k^{2}+3)(k^{2}+7)=k^{2}\times k^{2}+k^{2}\times7 + 3\times k^{2}+3\times7$.

Step3: Simplify terms

$k^{2}\times k^{2}=k^{4}$, $k^{2}\times7 = 7k^{2}$, $3\times k^{2}=3k^{2}$, and $3\times7 = 21$. Then $A=k^{4}+(7k^{2}+3k^{2})+21$.

Step4: Combine like - terms

$7k^{2}+3k^{2}=10k^{2}$, so $A = k^{4}+10k^{2}+21$.

Answer:

$k^{4}+10k^{2}+21$