Sure, let's factorize the given polynomial step by step.
Step 1: Combine like terms
Our polynomial is 7q^4 + 14q^2 - 112q^2 - 224. We have two terms involving q^2 that can be combined. Combining them yields
7q^4 - 98q^2 - 224.
Step 2: Common factor
Look for common factors in each term. The terms 7q^4, -98q^2, and -224 all share a common factor of 7. Factoring this out gives
7(q^4 - 14q^2 - 32).
Step 3: Factor Quadratic Expression
Now the polynomial is in the form of a quadratic expression: 7(q^2)^2 - 14q^2 - 32. To factorize it, find the two numbers that multiply together to give -32 (c term) and add together to give -14 (b term).
Those numbers are -16 and 2. Replacing -14q^2 with -16q^2 + 2q^2 gives
7[(q^2 - 16)q^2 + 2].
Step 4: Factor AQ^4 + BQ^2
This can be written as the product of two factors if it is in the form (AQ^2 + B)(Q^2 + C). Here, recognize that q^4 is the square of q^2, then we can rewrite the equation again in the form (A'Q'^2 + B')(Q'^2 + C) treating q^2 like a variable Q'. So, that gives us
7[(q^2 - 4)(q^2 + 4) + 2].
Step 5: Rearrange
Finally, rearranging the terms, we get
7*(q - 4)*(q + 4)*(q^2 + 2) which is our factored polynomial.