factor. \\5z^2 - 23z - 10\\

QUESTION IMAGE

$factor. \\5z^2 - 23z - 10\\$

Question

factor.

\\5z^2 - 23z - 10\\

Explanation:

Response

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Identify coefficients of the quadratic expression

We begin with the quadratic expression:
\[5z^2 - 23z - 10\]
The coefficients are:

$a = 5$
$b = -23$
$c = -10$

Find two numbers that multiply to ac and add to b

We calculate the product $ac$:
\[ac = 5 \times (-10) = -50\]
We need to find two integers that multiply to $-50$ and add up to $b = -23$.
Let's test factors of $-50$:

$-25 \times 2 = -50$
$-25 + 2 = -23$

The two numbers are $-25$ and $2$.

Rewrite the middle term and factor by grouping

We split the middle term $-23z$ using $-25z$ and $2z$:
\[5z^2 - 25z + 2z - 10\]
Now, we group the terms into two pairs:
\[(5z^2 - 25z) + (2z - 10)\]
Factor out the greatest common factor (GCF) from each group:

From $5z^2 - 25z$, factor out $5z$: $5z(z - 5)$
From $2z - 10$, factor out $2$: $2(z - 5)$

This gives:
\[5z(z - 5) + 2(z - 5)\]

Factor out the common binomial

Both terms share a common binomial factor of $(z - 5)$. Factoring it out yields:
\[(5z + 2)(z - 5)\]
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<answer>
Factor.
$5z^2 - 23z - 10$ = <blank>$(5z + 2)(z - 5)$</blank>
</answer>

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Answer: