Expand the following product. \[ \left(x^{2}-x-1\right)\left(x^{2}+7 x+12\right)= \]

Step 1 ：We are given the product of two polynomials: \(\left(x^{2}-x-1\right)\left(x^{2}+7 x+12\right)\).

Step 2 ：We can expand this product by applying the distributive property, also known as the FOIL method.

Step 3 ：The FOIL method stands for First, Outer, Inner, and Last. It is a way for remembering how to multiply two binomials. However, in this case, we are dealing with two trinomials, but the principle is the same.

Step 4 ：We will multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.

Step 5 ：After performing these operations, we obtain the expanded form of the product: \(x^{4} + 6x^{3} + 4x^{2} - 19x - 12\).

Step 6 ：Final Answer: The expanded form of the product \(\left(x^{2}-x-1\right)\left(x^{2}+7 x+12\right)\) is \(\boxed{x^{4} + 6x^{3} + 4x^{2} - 19x - 12}\).