We store the definition: $g\left(x\right)=x^2+5x$

Applying the definitions, $g\left(-2\right)+2$ becomes ${\left(-2\right)}^2+5\times \left(-2\right)+2$

Let's calculate: ${\left(-2\right)}^2+5\times \left(-2\right)+2$

ⓘ Powers are computed before multiplications, additions and subtractions. With a positive integer exponent, a power corresponds to a repeated product: ${\left(-2\right)}^2$ = $\left(-2\right)\times \left(-2\right)$ = $4$■Perform multiplications before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative: $5\times \left(-2\right)$ = $-10$

$4-10+2$

ⓘ Start from $4$ and move back $10$ on the number line, going below zero: $4-10$ = $-6$

$-6+2$

ⓘ Start from $-6$ and move forward $2$ on the number line, getting closer to zero: $-6+2$ = $-4$

▶Result:   $-4$