Write as a Set of Linear Factors f(x)=5x^4-4x^3+19x^2-16x-4

$f (x) = 5 x^{4} - 4 x^{3} + 19 x^{2} - 16 x - 4$

Regroup terms.

$f (x) = - 4 x^{3} - 16 x + 5 x^{4} + 19 x^{2} - 4$

Factor $- 4 x$ out of $- 4 x^{3} - 16 x$ .

Factor $- 4 x$ out of $- 4 x^{3}$ .

$f (x) = - 4 x \cdot x^{2} - 16 x + 5 x^{4} + 19 x^{2} - 4$

Factor $- 4 x$ out of $- 16 x$ .

$f (x) = - 4 x \cdot x^{2} - 4 x \cdot 4 + 5 x^{4} + 19 x^{2} - 4$

Factor $- 4 x$ out of $- 4 x (x^{2}) - 4 x (4)$ .

$f (x) = - 4 x (x^{2} + 4) + 5 x^{4} + 19 x^{2} - 4$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f (x) = - 4 x (x^{2} + 4) + 5 {(x^{2})}^{2} + 19 x^{2} - 4$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$f (x) = - 4 x (x^{2} + 4) + 5 u^{2} + 19 u - 4$

Factor by grouping.

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 4 = - 20$ and whose sum is $i = 19$ .

Factor $19$ out of $19 u$ .

$f (x) = - 4 x (x^{2} + 4) + 5 u^{2} + 19 (u) - 4$

Rewrite $19$ as $- 1$ plus $20$

$f (x) = - 4 x (x^{2} + 4) + 5 u^{2} + (- 1 + 20) u - 4$

Apply the distributive property.

$f (x) = - 4 x (x^{2} + 4) + 5 u^{2} - 1 u + 20 u - 4$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = - 4 x (x^{2} + 4) + 5 u^{2} - 1 u + 20 u - 4$

Factor out the greatest common factor (GCF) from each group.

$f (x) = - 4 x (x^{2} + 4) + u (5 u - 1) + 4 (5 u - 1)$

Factor the polynomial by factoring out the greatest common factor, $5 u - 1$ .

$f (x) = - 4 x (x^{2} + 4) + (5 u - 1) (u + 4)$

Replace all occurrences of $u$ with $x^{2}$ .

$f (x) = - 4 x (x^{2} + 4) + (5 x^{2} - 1) (x^{2} + 4)$

Factor $x^{2} + 4$ out of $- 4 x (x^{2} + 4) + (5 x^{2} - 1) (x^{2} + 4)$ .

Factor $x^{2} + 4$ out of $- 4 x (x^{2} + 4)$ .

$f (x) = (x^{2} + 4) (- 4 x) + (5 x^{2} - 1) (x^{2} + 4)$

Factor $x^{2} + 4$ out of $(5 x^{2} - 1) (x^{2} + 4)$ .

$f (x) = (x^{2} + 4) (- 4 x) + (x^{2} + 4) (5 x^{2} - 1)$

Factor $x^{2} + 4$ out of $(x^{2} + 4) (- 4 x) + (x^{2} + 4) (5 x^{2} - 1)$ .

$f (x) = (x^{2} + 4) (- 4 x + 5 x^{2} - 1)$

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$f (x) = (x^{2} + 4) (- 4 u + 5 u^{2} - 1)$

Factor by grouping.

Reorder terms.

$f (x) = (x^{2} + 4) (5 u^{2} - 4 u - 1)$

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 1 = - 5$ and whose sum is $i = - 4$ .

Factor $- 4$ out of $- 4 u$ .

$f (x) = (x^{2} + 4) (5 u^{2} - 4 u - 1)$

Rewrite $- 4$ as $1$ plus $- 5$

$f (x) = (x^{2} + 4) (5 u^{2} + (1 - 5) u - 1)$

Apply the distributive property.

$f (x) = (x^{2} + 4) (5 u^{2} + 1 u - 5 u - 1)$

Multiply $u$ by $1$ .

$f (x) = (x^{2} + 4) (5 u^{2} + u - 5 u - 1)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = (x^{2} + 4) ((5 u^{2} + u) - 5 u - 1)$

Factor out the greatest common factor (GCF) from each group.

$f (x) = (x^{2} + 4) (u (5 u + 1) - (5 u + 1))$

Factor the polynomial by factoring out the greatest common factor, $5 u + 1$ .

$f (x) = (x^{2} + 4) ((5 u + 1) (u - 1))$

Factor.

Replace all occurrences of $u$ with $x$ .

$f (x) = (x^{2} + 4) ((5 x + 1) (x - 1))$

Remove unnecessary parentheses.

$f (x) = (x^{2} + 4) (5 x + 1) (x - 1)$

Write as a Set of Linear Factors f(x)=5x^4-4x^3+19x^2-16x-4

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