Question

a rectangle has a height of $4a^2$ and a width of $6a^2 + 9a + 3$. express the area of the entire rectangle. your answer should be a polynomial in standard form. $6a^2 + 9a + 3$ (with a diagram of three rectangles labeled $4a^2$ on the side and $6a^2$, $9a$, $3$ on the top) area =

Explanation:

Step1: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by the product of its height and width. So, \( A=\text{height}\times\text{width} \). Here, the height is \( 4a^{2} \) and the width is \( 6a^{2}+9a + 3 \). So we need to compute \( 4a^{2}(6a^{2}+9a + 3) \).

Step2: Use the distributive property (multiplication over addition)

The distributive property states that \( c(a + b + d)=ca+cb+cd \). Applying this to \( 4a^{2}(6a^{2}+9a + 3) \), we get:
\( 4a^{2}\times6a^{2}+4a^{2}\times9a+4a^{2}\times3 \)

Step3: Simplify each term

• For the first term: \( 4a^{2}\times6a^{2}=(4\times6)a^{2 + 2}=24a^{4} \) (using the rule \( a^{m}\times a^{n}=a^{m + n} \))
• For the second term: \( 4a^{2}\times9a=(4\times9)a^{2+1}=36a^{3} \)
• For the third term: \( 4a^{2}\times3=(4\times3)a^{2}=12a^{2} \)

Step4: Combine the simplified terms

Combining the terms \( 24a^{4}+36a^{3}+12a^{2} \), we get the polynomial in standard form (where the exponents of \( a \) are in descending order).

Answer:

\( 24a^{4}+36a^{3}+12a^{2} \)