1. Statement of the problem: Solve the quadratic equation $7x^2+23x=60$.\n 2. Write the equation in standard form by moving all terms to one side: $$7x^2+23x-60=0$$.\n 3. Formula and method: For a quadratic $$ax^2+bx+c=0$$ we look for two numbers whose product is $a c$ and whose sum is $b$.\n 4. Compute $a c$: here $a=7$, $c=-60$ so $$a c = 7 \cdot (-60) = -420$$.\n 5. Find two integers with product $-420$ and sum $23$.\n 6. Test factor pairs and find $35$ and $-12$ because $35 \cdot (-12) = -420$ and $35 + (-12) = 23$.\n 7. Split the middle term using these numbers: $$7x^2 + 35x -12x -60 = 0$$.\n 8. Factor by grouping: $$7x(x+5) -12(x+5) = 0$$.\n 9. Factor out the common binomial to obtain $$(7x-12)(x+5)=0$$.\n 10. Solve each factor: $$7x-12=0\implies x=\frac{12}{7}$$\n $$x+5=0\implies x=-5$$.\n 11. Final answer: $x=\frac{12}{7}$ and $x=-5$.\n