If (t+2) units and `(t^(2) + 4)` units are length and breadth of a rectangle respectively, then find its area (in sq. units).

Options

A. `t^(3) + 2t^(2) + 4t + 8`
B. `t^(3) + t + 4t^(2) + 8`
C. `t^(3) + 8`
D. `t^(3) + t^(2) + 4 + 8t`

Answer

To find the area of a rectangle, we use the formula:

\[ \text{Area} = \text{Length} \times \text{Breadth} \]

Given:
- Length \( L = t + 2 \) units
- Breadth \( B = t^2 + 4 \) units

Now, we can substitute these values into the area formula:

\[ \text{Area} = (t + 2) \times (t^2 + 4) \]

Next, we will expand this expression using the distributive property (also known as the FOIL method for binomials):

1. Multiply \( t \) by each term in the second bracket:
\[ t \times t^2 = t^3 \]
\[ t \times 4 = 4t \]

2. Now, multiply \( 2 \) by each term in the second bracket:
\[ 2 \times t^2 = 2t^2 \]
\[ 2 \times 4 = 8 \]

Now, we combine all these results together:

\[ \text{Area} = t^3 + 4t + 2t^2 + 8 \]

Next, we can rearrange the terms in standard polynomial form:

\[ \text{Area} = t^3 + 2t^2 + 4t + 8 \]

Thus, the area of the rectangle in square units is:

\[ \text{Area} = t^3 + 2t^2 + 4t + 8 \]