Question

write an expression for the shaded region.
4x
3x + 5
x x + 2
(hint: find the area of the larger rectangle and the smaller rectangle and subtract)
answer: ( ax^b + cx^d )
a = type your answer...
b = type your answer...
c = type your answer...
d = type your answer...

Explanation:

Step1: Area of larger rectangle

The larger rectangle has length \(4x\) and width \(x + 2\). The area of a rectangle is length times width, so the area \(A_{large}=4x(x + 2)\). Using the distributive property (FOIL for monomial and binomial), we get \(A_{large}=4x\cdot x+4x\cdot2 = 4x^{2}+8x\).

Step2: Area of smaller rectangle

The smaller rectangle has length \(3x + 5\) and width \(x\). The area \(A_{small}=x(3x + 5)\). Distributing \(x\), we get \(A_{small}=3x^{2}+5x\).

Step3: Area of shaded region

The shaded area is the area of the larger rectangle minus the area of the smaller rectangle. So \(A_{shaded}=A_{large}-A_{small}=(4x^{2}+8x)-(3x^{2}+5x)\).

Simplify the expression: Combine like terms. \(4x^{2}-3x^{2}=x^{2}\) and \(8x - 5x = 3x\). Wait, no, wait. Wait, I made a mistake. Wait, the larger rectangle: length \(4x\), width \(x + 2\), so \(4x(x + 2)=4x^{2}+8x\). Smaller rectangle: length \(3x + 5\), width \(x\), so \(x(3x + 5)=3x^{2}+5x\). Then subtracting: \((4x^{2}+8x)-(3x^{2}+5x)=4x^{2}-3x^{2}+8x - 5x=x^{2}+3x\). Wait, but the answer format is \(ax^{b}+cx^{d}\). So comparing \(x^{2}+3x\) with \(ax^{b}+cx^{d}\), we have \(a = 1\), \(b = 2\), \(c = 3\), \(d = 1\). Wait, but let's re - check the area calculations.

Wait, maybe I messed up the larger rectangle's dimensions. Wait, the larger rectangle: top side is \(4x\), right side is \(x + 2\), so length is \(4x\), width is \(x + 2\), correct. Smaller rectangle: top side \(3x + 5\), right side \(x\), so length \(3x + 5\), width \(x\), correct. Then shaded area is large - small: \(4x(x + 2)-x(3x + 5)\). Let's expand again:

\(4x(x + 2)=4x^{2}+8x\)

\(x(3x + 5)=3x^{2}+5x\)

Subtract: \(4x^{2}+8x-3x^{2}-5x=(4x^{2}-3x^{2})+(8x - 5x)=x^{2}+3x\)

So in the form \(ax^{b}+cx^{d}\), \(a = 1\), \(b = 2\), \(c = 3\), \(d = 1\).

Answer:

\(a = 1\), \(b = 2\), \(c = 3\), \(d = 1\)