To fully factorize the expression [tex]\( 5x - 16x - 10 + 6x^2 \)[/tex], follow these steps:
1. Simplify the Expression:
Combine like terms in the polynomial.
[tex]\[
5x - 16x - 10 + 6x^2
\][/tex]
First, combine the [tex]\(x\)[/tex]-terms:
[tex]\[
5x - 16x = -11x
\][/tex]
So, the expression simplifies to:
[tex]\[
6x^2 - 11x - 10
\][/tex]
2. Factorize the Quadratic Expression:
We now need to factorize the quadratic polynomial [tex]\( 6x^2 - 11x - 10 \)[/tex].
To do this, we look for two binomials of the form [tex]\( (ax + b)(cx + d) \)[/tex] that multiply to give [tex]\( 6x^2 - 11x - 10 \)[/tex].
Notice if we write:
[tex]\[
(2x - 5)(3x + 2)
\][/tex]
We can verify this by expanding the binomials:
[tex]\[
(2x - 5)(3x + 2) = 2x \cdot 3x + 2x \cdot 2 - 5 \cdot 3x - 5 \cdot 2 = 6x^2 + 4x - 15x - 10 = 6x^2 - 11x - 10
\][/tex]
This matches our simplified quadratic expression.
Therefore, the fully factorized form of the expression [tex]\( 5x - 16x - 10 + 6x^2 \)[/tex] is:
[tex]\[
(2x - 5)(3x + 2)
\][/tex]
