study the products shown. is there a pattern?\n$(x + 3)^2 = x^2 + 6x + 9$\n$(x + 4)^2 = x^2 + 8x + 16$\n$(x…

Explanation:

Step1: Analyze the general form

The left - hand side of each equation is of the form ((x + n)^{2}), where (n) takes the values (3,4,5,6,\cdots)

Step2: Expand ((x + n)^{2}) using the formula ((a + b)^{2}=a^{2}+2ab + b^{2})

For ((x + n)^{2}), using the formula ((a + b)^{2}=a^{2}+2ab + b^{2}) with (a = x) and (b=n), we get ((x + n)^{2}=x^{2}+2nx + n^{2})

Step3: Compare with given examples

• For ((x + 3)^{2}), (n = 3), then (2nx=2\times3\times x = 6x) and (n^{2}=3^{2}=9), so ((x + 3)^{2}=x^{2}+6x + 9)
• For ((x + 4)^{2}), (n = 4), then (2nx=2\times4\times x=8x) and (n^{2}=4^{2} = 16), so ((x + 4)^{2}=x^{2}+8x + 16)
• For ((x + 5)^{2}), (n = 5), then (2nx=2\times5\times x = 10x) and (n^{2}=5^{2}=25), so ((x + 5)^{2}=x^{2}+10x + 25)
• For ((x + 6)^{2}), (n = 6), then (2nx=2\times6\times x=12x) and (n^{2}=6^{2}=36), so ((x + 6)^{2}=x^{2}+12x + 36)

The pattern is that for ((x + n)^{2}) (where (n) is a positive integer starting from 3 in the given examples), the expansion follows the formula ((x + n)^{2}=x^{2}+2nx + n^{2}), where the middle term's coefficient is (2n) (twice the constant term in the binomial) and the constant term is (n^{2}) (the square of the constant term in the binomial).

Answer:

The pattern is that for ((x + n)^{2}) (where (n) is a positive integer), ((x + n)^{2}=x^{2}+2nx + n^{2}) (the middle term coefficient is (2n) and the constant term is (n^{2}))