Factor the expression by grouping. First, the expression needs to be rewritten as $25a^{2}+pa+qa+12$. To find $p$ and $q$, set up a system to be solved.
$$p+q=-35$$ $$pq=25\times 12=300$$
Since $pq$ is positive, $p$ and $q$ have the same sign. Since $p+q$ is negative, $p$ and $q$ are both negative. List all such integer pairs that give product $300$.
Rewrite $25a^{2}-35a+12$ as $\left(25a^{2}-20a\right)+\left(-15a+12\right)$.
$$\left(25a^{2}-20a\right)+\left(-15a+12\right)$$
Factor out $5a$ in the first and $-3$ in the second group.
$$5a\left(5a-4\right)-3\left(5a-4\right)$$
Factor out common term $5a-4$ by using distributive property.
$$\left(5a-4\right)\left(5a-3\right)$$
Steps Using the Quadratic Formula
Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$25a^{2}-35a+12=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.
Now solve the equation $a=\frac{35±5}{50}$ when $±$ is plus. Add $35$ to $5$.
$$a=\frac{40}{50}$$
Reduce the fraction $\frac{40}{50}$ to lowest terms by extracting and canceling out $10$.
$$a=\frac{4}{5}$$
Now solve the equation $a=\frac{35±5}{50}$ when $±$ is minus. Subtract $5$ from $35$.
$$a=\frac{30}{50}$$
Reduce the fraction $\frac{30}{50}$ to lowest terms by extracting and canceling out $10$.
$$a=\frac{3}{5}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{4}{5}$ for $x_{1}$ and $\frac{3}{5}$ for $x_{2}$.
Multiply $\frac{5a-4}{5}$ times $\frac{5a-3}{5}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
