Expand the expression to a polynomial in standard form: \[ \left(2 x^{2}+3 x+2\right)\left(2 x^{2}-3 x+6\right) \]

To expand the expression \((2 x^{2}+3 x+2)(2 x^{2}-3 x+6)\), we apply the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial. 1. First, we multiply \(2x^2\) by each term in the second polynomial: \[ 2x^2 \cdot 2x^2 = 4x^4 \] \[ 2x^2 \cdot (-3x) = -6x^3 \] \[ 2x^2 \cdot 6 = 12x^2 \] 2. Next, we multiply \(3x\) by each term in the second polynomial: \[ 3x \cdot 2x^2 = 6x^3 \] \[ 3x \cdot (-3x) = -9x^2 \] \[ 3x \cdot 6 = 18x \] 3. Now, we multiply \(2\) by each term in the second polynomial: \[ 2 \cdot 2x^2 = 4x^2 \] \[ 2 \cdot (-3x) = -6x \] \[ 2 \cdot 6 = 12 \] Now we combine all these products: \[ 4x^4 + (-6x^3 + 6x^3) + (12x^2 - 9x^2 + 4x^2) + (18x - 6x) + 12 \] Combining like terms, we get: \[ 4x^4 + 0x^3 + (12 - 9 + 4)x^2 + (18 - 6)x + 12 \] \[ = 4x^4 + 0x^3 + 7x^2 + 12x + 12 \] Thus, the polynomial in standard form is: \[ \boxed{4x^4 + 7x^2 + 12x + 12} \]