Apply the FOIL method: multiply the first terms, outer terms, inner terms, and last terms. This gives x*x = x^2, x*3 = 3x, 2*x = 2x, and 2*3 = 6. Combining like terms results in x^2 + 5x + 6.

The degree of a polynomial is the highest power of the variable present. In 3x^3 + 2x^2 - x + 7, the highest exponent is 3. Therefore, the degree is 3.

When multiplying monomials, multiply the coefficients and add the exponents of like bases. Here, 5 multiplied by 3 gives 15 and x^2 multiplied by x^3 gives x^(2+3) or x^5. Hence, the product is 15x^5.

Use the distributive property (FOIL) to multiply the binomials: 2x*x gives 2x^2, 2x*4 gives 8x, -3*x gives -3x, and -3*4 gives -12. Combining 8x and -3x results in 5x. Thus, the expanded form is 2x^2 + 5x - 12.

The expression (x - 1)(x - 1) is equivalent to (x - 1)^2. Expanding a squared binomial gives x^2 - 2x + 1. This is the correct simplified form.

Multiply x by each term in (x^2 - x + 4) to get x^3 - x^2 + 4x, and 2 by each term to get 2x^2 - 2x + 8. Combining like terms yields x^3 + ( - x^2 + 2x^2) + (4x - 2x) + 8, which simplifies to x^3 + x^2 + 2x + 8.

The equation factors neatly into (x - 2)(x - 3) = 0. Setting each factor equal to zero gives x = 2 and x = 3. These are the solutions of the quadratic equation.

Distribute the constant 3 to each term inside the parentheses: 3*2x^2 gives 6x^2, 3*(-4x) yields -12x, and 3*5 produces 15. The expanded expression is 6x^2 - 12x + 15.

Using the quadratic formula, the discriminant is calculated as 3^2 - 4(2)(-5) = 9 + 40 = 49. This gives solutions x = (-3 + 7)/4 = 1 and x = (-3 - 7)/4 = -5/2. Thus, the correct answers are x = 1 and x = -5/2.

Search for two binomials that multiply to give 2x^2 + 7x + 3. Testing factors reveals that (2x + 1)(x + 3) expands correctly to 2x^2 + 7x + 3. Therefore, this is the proper factorization.

Multiply the terms using FOIL: 3x*2x gives 6x^2, 3x*5 gives 15x, -2*2x gives -4x, and -2*5 results in -10. Combining the middle terms, 15x - 4x equals 11x, so the final expression is 6x^2 + 11x - 10.

The quadratic factors into (x + 5)(x - 1) = 0. Setting each factor equal to zero gives the solutions x = -5 and x = 1. These are the correct roots of the equation.

Distribute each term of f(x) to every term in g(x): multiply x^2, -3x, and 2 by both x and 4. Combining like terms results in the cubic polynomial x^3 + x^2 - 10x + 8. This is the simplified product of the two functions.

Since (x - 2) is a factor, divide the cubic polynomial by (x - 2) to obtain a quadratic factor. The resulting quadratic factors as (x - 3)(x + 1). Setting each factor to zero provides the solutions x = 2, x = 3, and x = -1.

The expression 4x^2 - 12x + 9 is recognized as a perfect square trinomial. It factors into (2x - 3)^2 because (2x - 3)(2x - 3) expands back to the original expression. This is the complete factorization.

Multiply each term from the first polynomial by every term in the second: 2x^2*x^2 gives 2x^4, 2x^2*x gives 2x^3, and 2x^2*(-2) gives -4x^2. Continuing with -3x and 4, and combining like terms results in 2x^4 - x^3 - 3x^2 + 10x - 8.

Let y = x^2, which turns the equation into y^2 - 5y + 4 = 0. Factoring yields (y - 1)(y - 4) = 0, so y = 1 or y = 4. Re-substitute to find x: from x^2 = 1 you get x = ±1, and from x^2 = 4 you get x = ±2.

Using synthetic division with the value 1 (from the factor x - 1), bring down the first coefficient 2. Proceeding through the synthetic division steps yields quotient coefficients 2, 5, and a remainder of 6. Thus, the division results in 2x^2 + 5x with a remainder of 6.

Since (x - 3) is a known factor, divide the polynomial by (x - 3) to obtain the quadratic factor. The resulting quadratic, x^2 + x - 2, further factors into (x + 2)(x - 1). Therefore, the complete factorization is (x - 3)(x + 2)(x - 1).

Rewrite the left side as ((x - 2)(x + 2))^2 which is (x - 2)^2(x + 2)^2. Cancelling the common term (x - 2)^2 (while checking the case x = 2 separately) leads to (x + 2)^2 = 9. Taking square roots gives x + 2 = ±3, resulting in x = 1 or x = -5, and including the case x = 2, the solutions are x = -5, 1, and 2.