Expand the expression to a polynomial in standard form \[ \left(2 x^{2}+x+2\right)\left(x^{2}-7 x-6\right) \]

To expand the expression \((2x^{2} + x + 2)(x^{2} - 7x - 6)\), we'll use the distributive property (also known as the FOIL method for binomials), applying it to each term in the first polynomial to each term in the second polynomial. 1. First, distribute \(2x^2\): \[ 2x^2 \cdot (x^2 - 7x - 6) = 2x^4 - 14x^3 - 12x^2 \] 2. Next, distribute \(x\): \[ x \cdot (x^2 - 7x - 6) = x^3 - 7x^2 - 6x \] 3. Lastly, distribute \(2\): \[ 2 \cdot (x^2 - 7x - 6) = 2x^2 - 14x - 12 \] Now, we combine all the terms we have obtained: \[ 2x^4 - 14x^3 - 12x^2 + x^3 - 7x^2 - 6x + 2x^2 - 14x - 12 \] Next, we combine like terms: - For \(x^4\): \(2x^4\) - For \(x^3\): \(-14x^3 + x^3 = -13x^3\) - For \(x^2\): \(-12x^2 - 7x^2 + 2x^2 = -17x^2\) - For \(x\): \(-6x - 14x = -20x\) - Constant term: \(-12\) Putting this all together, we have the expanded expression in standard form: \[ \boxed{2x^4 - 13x^3 - 17x^2 - 20x - 12} \]