Verify and expand the identity: $(x + 1) (x + 4)$ geometrically.

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Solution

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, $A B C D =$ $(x + 1) (x + 4)$
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 $=$ $(x)^{2}$ and the area of yellow rectangle $=$ length $\times$ breadth $=$ $4 x$
Step 6: Area of blue rectangle $=$ $x$ and the area of green rectangle $=$ $4$
Step 7: Area of full rectangle $=$ area of pink square $+$ area of yellow rectangle $+$ area of blue rectangle $+$ area of green rectangle.
i.e., $(x + 1) (x + 4) = (x)^{2} + 4 x + x + 4$
$(x + 1) (x + 4) = (x)^{2} + x (4 + 1) + 4$
Hence, geometrically we expanded the identity $(x + 1) (x + 4) = (x)^{2} + x (4 + 1) + 4$ .

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