## Expanding (9x+1)^2

The expression (9x+1)^2 represents the square of the binomial (9x+1). To expand this expression, we can use the **FOIL method** or the **square of a binomial formula**.

### FOIL Method

FOIL stands for **First, Outer, Inner, Last**. This method involves multiplying each term of the first binomial by each term of the second binomial.

**First:**Multiply the first terms of each binomial: 9x * 9x = 81x^2**Outer:**Multiply the outer terms of the binomials: 9x * 1 = 9x**Inner:**Multiply the inner terms of the binomials: 1 * 9x = 9x**Last:**Multiply the last terms of each binomial: 1 * 1 = 1

Now we add all the terms together: 81x^2 + 9x + 9x + 1

Finally, we combine the like terms: **81x^2 + 18x + 1**

### Square of a Binomial Formula

The square of a binomial formula states: (a + b)^2 = a^2 + 2ab + b^2

Applying this to our problem:

- a = 9x
- b = 1

Substituting these values into the formula:

(9x + 1)^2 = (9x)^2 + 2(9x)(1) + (1)^2

Simplifying:

(9x + 1)^2 = 81x^2 + 18x + 1

### Conclusion

Both the FOIL method and the square of a binomial formula lead to the same expanded expression: **81x^2 + 18x + 1**.

This expanded form is essential for solving equations, simplifying expressions, and performing other algebraic operations involving (9x+1)^2.