Answer :
We need to factor the expression
[tex]$$
x^4 + 4x^2 - 45.
$$[/tex]
A useful strategy is to treat this as a quadratic in terms of [tex]$x^2$[/tex]. Follow these steps:
1. Let
[tex]$$
u = x^2.
$$[/tex]
Then the expression becomes:
[tex]$$
u^2 + 4u - 45.
$$[/tex]
2. Now, we need to factor the quadratic [tex]$u^2 + 4u - 45$[/tex]. We look for two numbers that multiply to [tex]$-45$[/tex] and add to [tex]$4$[/tex]. These numbers are [tex]$9$[/tex] and [tex]$-5$[/tex] because:
[tex]$$
9 \cdot (-5) = -45 \quad \text{and} \quad 9 + (-5) = 4.
$$[/tex]
3. Using these numbers, we factor the quadratic as:
[tex]$$
u^2 + 4u - 45 = (u - 5)(u + 9).
$$[/tex]
4. Substitute back [tex]$x^2$[/tex] for [tex]$u$[/tex]:
[tex]$$
(u - 5)(u + 9) = (x^2 - 5)(x^2 + 9).
$$[/tex]
Thus, the complete factorization of the original expression is:
[tex]$$
(x^2 - 5)(x^2 + 9).
$$[/tex]
[tex]$$
x^4 + 4x^2 - 45.
$$[/tex]
A useful strategy is to treat this as a quadratic in terms of [tex]$x^2$[/tex]. Follow these steps:
1. Let
[tex]$$
u = x^2.
$$[/tex]
Then the expression becomes:
[tex]$$
u^2 + 4u - 45.
$$[/tex]
2. Now, we need to factor the quadratic [tex]$u^2 + 4u - 45$[/tex]. We look for two numbers that multiply to [tex]$-45$[/tex] and add to [tex]$4$[/tex]. These numbers are [tex]$9$[/tex] and [tex]$-5$[/tex] because:
[tex]$$
9 \cdot (-5) = -45 \quad \text{and} \quad 9 + (-5) = 4.
$$[/tex]
3. Using these numbers, we factor the quadratic as:
[tex]$$
u^2 + 4u - 45 = (u - 5)(u + 9).
$$[/tex]
4. Substitute back [tex]$x^2$[/tex] for [tex]$u$[/tex]:
[tex]$$
(u - 5)(u + 9) = (x^2 - 5)(x^2 + 9).
$$[/tex]
Thus, the complete factorization of the original expression is:
[tex]$$
(x^2 - 5)(x^2 + 9).
$$[/tex]
