Solve the Quadratic Equation: 5x²+15x-40=3x+x²

Solve the following problem:

_{$5x^2+15x-40=3x+x^2$}

Solve the given equation:

$5x^2+15x-40=3x+x^2$

$$Organize the equation by moving and combining like terms:

$5x^2+15x-40=3x+x^2 \\ 5x^2+15x-40-3x-x^2=0 \\ 4x^2+12x-40=0$

Note that all coefficients as well as the free term are multiples of 4, hence we'll divide both sides of the equation by 4:

$4x^2+12x-40=0\hspace{6pt}\text{/}:4 \\ x^2+3x-10=0$

Now we notice that the coefficient of the squared term is 1, therefore, we can (try to) factor the expression on the left side using quick trinomial factoring:

Let's look for a pair of numbers whose product equals the free term in the expression, and whose sum equals the coefficient of the first-degree term, meaning two numbers $m,\hspace{2pt}n$ that satisfy the desired values:

$m\cdot n=-10\\ m+n=3\\$From the first requirement mentioned, that is - from the multiplication, we notice that the product of the numbers we're looking for must be negative. Therefore we can conclude that the two numbers have different signs, according to multiplication rules. Remember that the possible factors of 10 are 5 and 2 or 10 and 1, fulfilling the second requirement mentioned. This together with the fact that the numbers we're looking for have different signs leads us to the conclusion that the only possibility for the two numbers we're looking for is:

$\begin{cases} m=5 \\ n=-2 \end{cases}$

Therefore we'll factor the expression on the left side of the equation to:

$x^2+3x-10=0 \\ \downarrow\\ (x+5)(x-2)=0$

Remember that the product of expressions equals 0 only if at least one of the multiplied expressions equals zero,

Therefore we'll obtain two simple equations and solve them by isolating the variable in each:

$x+5=0\\ \boxed{x=-5}$

or:

$x-2=0\\ \boxed{x=2}$

Let's summarize the solution of the equation:

$5x^2+15x-40=3x+x^2 \\ 4x^2+12x-40=0\\ x^2+3x-10=0 \\ \downarrow\\ (x+5)(x-2)=0 \\ \downarrow\\ x+5=0\rightarrow\boxed{x=-5}\\ x-2=0\rightarrow\boxed{x=2}\\ \downarrow\\ \boxed{x=-5,2}$

Therefore the correct answer is answer A.