Question

a rectangle has a height of (3n^2) and a width of (4n^2 - 5n - 1). express the area of the entire rectangle. your answer should be a polynomial in standard form. (4n^2 + -5n + -1) diagram of three colored rectangles with height (3n^2) on the left area = box

Explanation:

Step1: Recall the formula for the area of a rectangle

The area \( A \) of a rectangle is given by the product of its height \( h \) and width \( w \), i.e., \( A = h \times w \). Here, \( h = 3n^2 \) and \( w = 4n^2 - 5n - 1 \). So we need to compute \( 3n^2 \times (4n^2 - 5n - 1) \).

Step2: Apply the distributive property (multiplication over addition/subtraction)

Using the distributive property \( a(b + c + d)=ab+ac + ad \), we multiply \( 3n^2 \) with each term in the width polynomial:

• Multiply \( 3n^2 \) by \( 4n^2 \): \( 3n^2\times4n^2 = 12n^{2 + 2}=12n^4 \)
• Multiply \( 3n^2 \) by \( - 5n \): \( 3n^2\times(-5n)=-15n^{2+1}=-15n^3 \)
• Multiply \( 3n^2 \) by \( -1 \): \( 3n^2\times(-1)=-3n^2 \)

Step3: Combine the terms

Combine the results from the previous step: \( 12n^4-15n^3 - 3n^2 \)

Answer:

\( 12n^4 - 15n^3 - 3n^2 \)