Question

factor completely. $5z^{2}+14z + 9$

Explanation:

Step1: Multiply leading - coeff and constant

For the quadratic $5z^{2}+14z + 9$, $a = 5$, $b = 14$, $c = 9$. We calculate $a\times c=5\times9 = 45$.

Step2: Find two numbers for $ac$

We need two numbers that multiply to $45$ and add up to $14$. The numbers are $5$ and $9$ since $5\times9=45$ and $5 + 9=14$.

Step3: Rewrite middle term

Rewrite $14z$ as $5z+9z$. So, $5z^{2}+14z + 9=5z^{2}+5z+9z + 9$.

Step4: Group terms

Group the terms: $(5z^{2}+5z)+(9z + 9)$.

Step5: Factor out GCF from each group

From the first group $5z^{2}+5z$, the GCF is $5z$, so $5z^{2}+5z=5z(z + 1)$. From the second group $9z + 9$, the GCF is $9$, so $9z + 9=9(z + 1)$.

Step6: Factor out common binomial

We have $5z(z + 1)+9(z + 1)$. Factoring out the common binomial $(z + 1)$ gives $(z + 1)(5z+9)$.

Answer:

$(z + 1)(5z+9)$