Question

Explain and expand the identity geometrically: $(4x+1)(4x+5)$

Answer 1

Step 1: Draw a square and cut into $4$ parts.
Step 2: There are $4$ rectangle and $1$ square.
Step 3: Area of the full rectangle, $ABCD=$ $(4x+1)(4x+5)$
Step 4: Now we have to find the area of inside square and rectangle as shown in the figure.
Step 5: Consider the area of pink square $=$ $(4x{)}^2$ and the area of yellow rectangle $=$ length $\times$ breadth $=$ $20x$
Step 6: Area of blue rectangle $=$ $4x$ and the area of green rectangle $=$ $5$
Step 7: Area of full rectangle $=$ area of pink square $+$ area of yellow rectangle $+$ area of blue rectangle $+$ area of green rectangle.
i.e., $(4x+1)(4x+5)=(4x{)}^2+x+5x+5$
$(4x+1)(4x+5)=(4x{)}^2+x(1+5)+5$
Hence, geometrically we expanded the identity $(4x+1)(4x+5)=(4x{)}^2+x(1+5)+5$.