Factor the polynomial $x^4 - 4x^2 - 5$ by using substitution to rewrite it as a quadratic trinomial.

Step 1: Identify the relationship between the variable terms. The variable part of the middle term is $x^2$, and its square, $(x^2)^2 = x^4$, is the variable part of the first term. Step 2: Let $u = x^2$ and substitute $u$ into the trinomial to put it in standard form: $u^2 - 4u - 5$ Step 3: Factor the new trinomial in terms of $u$. Find two numbers that multiply to $-5$ and add to $-4$. These numbers are $-5$ and $1$: $(u + 1)(u - 5)$ Step 4: Replace $u$ with the original expression, $x^2$: $(x^2 + 1)(x^2 - 5)$ Step 5: Check the result by multiplying the factors: $(x^2 + 1)(x^2 - 5) = x^4 - 5x^2 + x^2 - 5 = x^4 - 4x^2 - 5$ ✓

The factored form is $(x^2 + 1)(x^2 - 5)$.