Is there a method to factor this equation?

Question

This equation appears in a question book but the method for factoring (if there is one) is not familiar to me.

$$x^3 + x^2 -4x -4 $$

The solution manual lists the following as the factorised form of the above equation:

$$x^2(x + 1) - 4(x + 1)$$

What is the method for achieving this? Thanks

Answer 1

This is just factoring out the common factor of $x^2$ from the first two terms, and the common factor of $-4$ from the last two terms. However, the correct factorization would be

$$x^2(x+1) - 4(x+1)$$

in that light. This suggests the notion of "factoring by grouping", if this is a keyword you want to look up further, since the next most reasonable factorization is factoring out the common factor of $x+1$:

$$(x^2-4)(x+1)$$

Khan Academy and this site discuss some examples in detail.

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Note 1: The original question statement used the factorization of $x^2(x+1) - 4(x-1)$, hence my comment on the factorization being wrong. The question has since been edited.

Note 2: As astutely brought up in the comments, the $x^2-4$ factor can be further factored as $(x-2)(x+2)$. This is often known as the "difference of squares" identity. Further discussion and applications are on Khan Academy, Brilliant, and Wikipedia.

Answer 2

$-4(x+1)$, you mean. You then factor out the $(x+1)$, obtaining $(x^2-4)(x+1)$. This is often called factoring by grouping.

Answer 3

Observe that for $p(x) = x^3+x^2 - 4x - 4$, then $p(-1) = 0$. Thus as a polynomial, $p(x)$ has a factor $x+1$. You can then use synthetic division to obtain the quotient and can factor $p(x)$ into a product of $x+1$ and a quadratic polynomial.