How do you factor the polynomial $f(x) = 3{x^2} + 16x - 35$?

Hint: This problem deals with factoring the given polynomial which is a quadratic polynomial. A quadratic polynomial is a polynomial of degree 2. Factoring the given quadratic polynomial is done by splitting the $x$ term into the factors of the product of the coefficient of the ${x^2}$ term and the constant.

Complete step-by-step solution:
Given there is a polynomial of degree two, which is a function of $x$, and we have to factorize the given polynomial.
The given polynomial is a quadratic polynomial of $x$.
Now consider the given quadratic polynomial as shown below:
$ \Rightarrow f(x) = 3{x^2} + 16x - 35$
Now multiply the coefficient of the ${x^2}$ term and the constant, which is equal to $3\left( { - 35} \right) = - 105$, so the product is equal to -105.
Now consider the obtained product which is equal to -105, and we have to split the product into two factors which can be the algebraic sum of the coefficient of the $x$ term.
Here the coefficient of the $x$ term is equal to 16, now the best two factors of the product of -105, which can be expressed as the algebraic sum of 16 are the factors 21 and -5.
So now the sum of the numbers 21 and -5 is exactly equal to 16.
Now expressing the quadratic polynomial in factored form as shown below:
 $ \Rightarrow f(x) = 3{x^2} + 16x - 35$
$ \Rightarrow f(x) = 3{x^2} + 21x - 5x - 35$
Now taking the term $3x$ common from the first two terms and the term 5 common from the last two terms, as shown below:
$ \Rightarrow f(x) = 3x\left( {x + 7} \right) - 5\left( {x + 7} \right)$
Now taking the term $\left( {x + 7} \right)$ from the above expression as shown below:
$ \Rightarrow f(x) = \left( {x + 7} \right)\left( {3x - 5} \right)$
The factors of the function $f(x) = 3{x^2} + 16x - 35$ are equal to $f(x) = \left( {x + 7} \right)\left( {3x - 5} \right)$.

Note: Please note that this problem can also be done in another method, which is by equating the given quadratic polynomial to zero, which then becomes a quadratic equation in the form of $a{x^2} + bx + c = 0$, we can find the roots of the quadratic equation with the formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, and then factorize the roots.