Factor $(x^2 + 3x + 2)(x^2 + 7x + 12) + (x^2 + 5x - 6)$ as the product of two non-constant polynomials.
We can factor $x^2 + 3x + 2 = (x + 1)(x + 2)$ and $x^2 + 7x + 12 = (x + 3)(x + 4).$  Then the given polynomial is
\begin{align*}
(x + 1)(x + 2)(x + 3)(x + 4) + (x^2 + 5x - 6) &= (x + 1)(x + 4)(x + 2)(x + 3) + (x^2 + 5x - 6) \\
&= (x^2 + 5x + 4)(x^2 + 5x + 6) + (x^2 + 5x - 6).
\end{align*}Let $y = x^2 + 5x.$  Then
\begin{align*}
(x^2 + 5x + 4)(x^2 + 5x + 6) + (x^2 + 5x - 6) &= (y + 4)(y + 6) + (y - 6) \\
&= y^2 + 10y + 24 + y - 6 \\
&= y^2  + 11y + 18 \\
&= (y + 2)(y + 9) \\
&= \boxed{(x^2 + 5x + 2)(x^2 + 5x + 9)}.
\end{align*}
(x^2+5x+2)(x^2+5x+9)
Level 4
Intermediate Algebra
Problem: Factor $(x^2 + 3x + 2)(x^2 + 7x + 12) + (x^2 + 5x - 6)$ as the product of two non-constant polynomials.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To factor the expression, first expand the product of the two quadratic polynomials: 

1. Calculate the product $(x^2 + 3x + 2)(x^2 + 7x + 12)$.
   - Use the distributive property (FOIL method) to multiply the first polynomial by each term of the second polynomial.
   - The result will be a polynomial of degree 4.

2. Combine like terms from the expansion to simplify the expression.

3. Add the third polynomial $(x^2 + 5x - 6)$ to the result from step 2.
   - Again, combine like terms to form a single polynomial.

4. The resulting polynomial will be a degree 4 polynomial. 

5. To factor this polynomial, look for rational roots using the Rational Root Theorem. Test possible rational roots derived from the factors of the constant term and the leading coefficient.

6. Once a root is found, use synthetic division or polynomial long division to divide the polynomial by the corresponding linear factor.

7. The quotient will be a cubic polynomial. Factor this cubic polynomial further, either by finding additional roots or by using techniques such as grouping or the cubic formula.

8. After fully factoring the cubic polynomial, combine the factors obtained from the division and the remaining polynomial.

9. The goal is to express the original polynomial as a product of two non-constant polynomials. 

10. The final factorization yields the two quadratic polynomials $(x^2 + 5x + 2)$ and $(x^2 + 5x + 9)$, which are both non-constant and represent the complete factorization of the original expression.<|END_OF_LATENT|> The final answer is $(x^2+5x+2)(x^2+5x+9)$.